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The Complexity of Knots
Quo Vadis, Graph Theory? J. Gimbel, J.W. Kennedy & L.V. Quintas (4s.) Annals of Discrete Mathematics, 55, 159-172 (1993) 0 1993 Elsevier Science Publishers B.V. All rights reserved.
THE COMPLEXITY OF KNOTS Dominic J.A. WELSH Merton College, University of Oxford Oxford, ENGLAND
Abstract The paper considers the computational complexity of classifying knots and of determining several well known knot invariants, both with and without an oracle for testing isotopy.
1.
Introduction
The title of this paper is taken from Tait’s seminal work on knots. Although it is unlikely that Tait was thinking of complexity in the sense that it is used here, the underlying problems encountered by Tait are basically the same questions that we shall be considering. Two fundamental problems of knot theory are:
Is a knotted curve really knotted? Are two knotted curves really the same knot? These were clearly the topics in Tait’s mind when he wrote [l] (p.300): “Before taking up the question of the complexity of a knot a word or two must be said about the methods of reducing any given knot to its
simplest form. I have not been able as yet to find any general method of doing this, nor have I even discovered what would probably solve this diflculty, any perfectly general method ofpronouncing at once from its scheme or otherwise whether a knot is reducible or not.” Tait might be amused to know that one hundred years later and despite a massive effort, these problems are still difficult. For example, until 1974 the knot diagrams shown in Figure 1 were wrongly thought to represent different knots.
Figure 1: In this paper we shall discuss various algorithmic and complexity questions arising from knots. Familiarity with concepts from combinatorics will be assumed; the knot theory concepts will be defined, more details may be found in the books of Burde and Zieschang [2] and Kauffman [3]. The complexity terminology follows Garey and Johnson [4].
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The complexity classes P, NP, #P and EXPTIME have their usual meanings and we remind the reader that although it is known that (1.1)
P G N P G P#'c EXPTIME,
the only inclusion which has been proved to be strict is that P i s distinct from EXPZ7ME. The notation A = B implies A is Turing reducible to B. 2.
Isotopy of Links
A link L with c(L) components in the three-sphere S3 is a smooth sub-manifold that consists of c(L) disjoint simple closed curves. A knot is a link with one component. Two links K, L are isotopic if there exists a homotopy h,: 9 +9 ( 0 I t I 1),such that h , = 1, each h, is a homeomorphism and h, ( K ) = L . We restrict attention to tame links and thus we may assume that, for each link considered, the projection x [ L ] of L to R2 is a finite 4-regular plane graph. The link diagram D(L) of L arising from x[L] is obtained by indicating at each crossing which of the two curve segments goes over the other.
The fundamental theorem of Reidemeister [5]states: Theorem 2.1:
Two links K and L are isotopic if and only if any link diagram of K can be transformed into any link diagram of L by a finite sequence of the moves (I), (II), (111) and their inverses.
Figure 2: Reidemeister moves. We will say that two link diagrams are equivalent if the corresponding links are isotopic. We denote this equivalence of link diagrams and isotopy of links by D -D' and K K'.
-
These moves, known as Reidemeister moves are applied locally. In each case, away from the crossings to which the move is being applied, the diagrams remain unchanged. It is important to note that there are other notions of equivalence; for example one could regard K1 and K2 as equivalent if there exists an autohomeomorphism of 9 which maps K1 to K2. This is a weaker notion than isotopy, for example taking K1 and K2 as shown in Figure 3; they are the right hand trefoil knot and its mirror image the left hand trefoil; these are equivalent in the above sense but are not isotopic; there is no sequence of Reidemeister moves which will transform K 1 to K2. In other words they are chiral. Henceforth we shall exclusively use equivalence in the sense of isotopic equivalence, and thus can rely entirely on the calculus of Reidemeister moves to demonstrate it.
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Figure 3: Two non-isotopic knots. A Graphic Interpretation of Link Isotopy
Given any link diagram D we note that if overlunder crossings are ignored it can be regarded as a 4-regular plane graph G(D).Accordingly it is Eulerian and its dual plane graph, consisting of the faces of G ( D )can be 2-coloured. We color the boundary faces black, and if two faces share a crossing we join them by a signed edge according to the convention shown in Figure 4.
+ve crossing
-ve crossing
Figure 4 In this way, given any link diagram D we get a plane signed graph S(D) in which each edge, appropriately signed, corresponds to a crossing in D.
Conversely, given any plane graph G with its edges signed + or -, we can associate with G, in a canonical way, a link diagram D(G) such that S(D(G))= G. The construction is easy; draw the medial graph of G , call it m(G) and this will be the knot diagram where the over/ under nature of the crossings is determined by the sign of the appropriate edge in G. We leave the details to the reader. From this correspondence between signed planar graphs and link diagrams it is straightforward to verify that the Reidemeister moves (I) -(HI) have the following interpretations as graph transformations Add or delete a loop or isthmus of any sign. (I*): (II*(a)): Insert or delete a parallel pair of oppositely signed edges between any pair of vertices, provided this does not contravene the planar representation. (II*(b)): Contract a pair of oppositely signed edges which are in series. Insert a pair of oppositely signed series edges at a vertex as shown in Figure 5.
(HI*):
The signed star triangle transformation (Figure 6).
Example: Consider the trefoil and its mirror image as shown in Figure 3. Their associated signed graphs D , D a r e shown below. How does one show that K1, K2 are not isotopic? Alternatively, how does one prove that there is no sequence of signed graph moves from D to D?
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Figure 5:
Figure 6 This difficulty is at the heart of the problems considered in this paper. It highlights a property of links which received much attention by Tait, who described K as amphicheiral, (now often called achiral) if K is isotopic to its mirror image. Deciding whether a knot is achiral is still difficult.
Figure 7: As another example of equivalence, take any plane graph G and let each edge be +ve, giving the signed graph G+. Now take the dual plane graph G* with each edge negatively signed, giving (@)-. Then it is not difficult to check
(2.2) The links having diagrams D(G+)and D[( @)-I are isotopic.
Moreover, we can show: (2.3)If the plane graph G has n edges then G+ may be transformed to (G*)- by O(n)Reidemeister moves.
This is just a special case of a more general result:
Result 2.4: For any plane graph G if S denotes a signed version of G, and Sdenotes its e a n a r dual G * with the corresponding edges oppositely signed then S can be transformed to S by O(IE{G)I) of the moves I*-HI*.
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Sketch of Proof: Given a link diagram D with n crossings, which represents the link L , then if A denotes any boundary face of D,and the arc PQ represents the common internal boundary of face A as illustrated in Figure 8 then the diagram fi can be transformed to the link diagram D'by a
Figure 8: sequence of O(n)Reidemeister moves corresponding to pulling PRQ through the diagram. It is now easy to check that the graph of D'is in fact the dual graph of D.We call this move the fold. For more on the relation between graphs and knots see [6] 3. The Complexity of Knot Triviality and Isotopy As mentioned earlier, one of the most fundamental algorithmic questions about knots is to decide whether a knot is trivial. In the now standard terminology of complexity questions we formalize this question as follows: KNOT TRIVIALITY:
Instance: A knot diagram D. Question: Is D topologically equivalent to the unknot? There is an involved and difficult algorithm due to Haken [71, see also Schubert [8] and Hemion [9], which shows that this problem is decidable. However, as far as I am aware, the status of this problem in the complexity hierarchy is not known. More precisely, we pose the fundamental problem: Problem3.1: Find a function f : Z + Z such that if D is a knot diagram which is equivalent to the unknot and which has n crossings then there exists a sequence of at mostf(n) Reidemeister moves which demonstrates the equivalence. More succinctly, we are looking for a bound on the length of Reidemeister proofs of equivalence to the unknot. An example of a knot diagram which in order to be shown equivalent to the unknot needs an increase in the number of crossings, is the diagram K shown in Figure 9. However, in a sense this highlights the inadequacy of the Reidemeister moves as a knot calculus since if one allows the fold as a single move, illustrated by the transition from K to k,then it is not so easy to find examples of diagrams with this property.
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Figure 9 This prompts the question:
Problem 3.2 Find a sequence of link diagrams D ieach representing the unknot and such that the number of crossings, nj, of D icannot be reduced to the unknot in fewer than n: Reidemeister moves. In a discussion with L. Kauffman, we decided that the following construction may provide a class of hard examples to unravel. Take a fairly complicated diagram of the unknot, having say m crossings. Now fold it in half, then fold it in half again and continue in this way for perhaps k folds. This will give a diagram D which still represents the unknot, has approximately q m k ) crossings, and would seem to demand very many Reidemeister moves in any unraveling process. However, proving this assertion seems very difficult. A closely related, but even harder problem, is the following: KNOT EQUIVALENCE:
Instance: Two knot diagrams D1, D2. Question: Do Dland 4 represent equivalent knots? It would be a major advance to show that KNOT EQUIVALENCE belonged to the complexity class EXPTZME; though again there is no firm evidence indicating that the problem is not in polynomial time P.
4. Classical Link Invariants Two link diagrams which have different numbers of components obviously represent non isotopic links. This is probably the simplest example of a link invariant and is used as a first subdivision of links in the classical tables of links (see for example [lo]). A less trivial invariant is the crossing number. This is defined to be the minimum number of crossings in any diagram representing the link. Determining the crossing number of a link is difficult. Expressed more formally, define the problem: CROSSING NUMBER:
Instance: A knot K represented by a diagram D , together with an integer k. Question: Is the crossing number of K < k? Since a knot has zero crossing number if and only if it is trivial, we have: (4.1) KNOT TRIVIALITY = CROSSING NUMBER.
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Tait introduced another knot invariant which is superficially related to crossing number and which he called beknottedness. It is the minimum number of changes of sign of crossings which reduce the knot to triviality. He remarks [l] (p.308) “There must be some very simple method of determining the amount of beknottedness for any given knot: but I have not hit upon it. ” He might have been surprised to know that determining the beknottedness of a knot is still an immensely difficult problem, even for diagrams with as few as 9 crossings, see for example [Ill. A link diagram D is alternating if and only if it has the property that in D the crossings are alternately overlunderlover.. .. While it is trivial to verify that a given diagram is alternating, it is highly nontrivial to test whether a given link L has a representation as an alternating link diagram. Any such link is called alternating. Equivalently, a link diagram D is alternating if and only if the associated signed graph has all its edges the same sign. This is easy to see. Thus, the question of deciding whether a link is alternating reduces to: Problem 4 3:
Is there a polynomial time algorithm which will check whether a given signed graph is transformable to a monosigned graph using the moves I*-HI*? If there were such an algorithm it would settle the knot triviality question, since a recent result of Murasugi [12] and Thistlethwaite [13], which settles one of the long standing Tait conjectures, states:
Theorem 43: If L is an alternating link and D is an alternating diagram representing L then provided D has no nugatory crossing it has a minimum number of crossings over all diagrams representing L. (A crossing is nugatory in D if, as is shown in Figure 10, it corresponds exactly to the corresponding edge of the graph G(D)being an isthmus).
Figure 10: A nugatory crossing. It is clear from this and the fact that nugatory crossings are easily recognized and eliminated, that finding an alternating representation of an alternating knot is at least as hard (in the computational sense) as deciding whether a knot is trivial. Hence we define the problem: ALTERNATING: Instance: A link diagram D. Question: Does D represent an alternating link?
We have immediately (4.4) KNOT TRIVIALITY = ALTERNATING.
One of the most powerful invariants of a knot is its group, or more precisely the funda-
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mental group of the knot complement, and it is natural to examine its complexity. The origin of this invariant can be traced back to Poincark. A series of results culminating in a recent
result of Gordon and Luecke [14] show that it is close to providing a complete classification of knots. First we observe:
(4.5)From a knot diagram it is easy to obtain in polynomial time a presentation of the knot group in terms of generators and relations. The well known Wirtinger presentation of an ncrossing diagram gives a presentation of n generators and n relations and can be found in time o(n2). (4.6) The famous theorem of Dehn and Papkyriakopoulos (1957) states that K has a group presented by a single generator if and only if K is the unknot. In other words:
(4.7) The trivial knot is completely characterized by its group. Unfortunately this characterization is not always easy to use and at the moment it still does not seem possible to use it to produce even an exponential time algorithm for deciding knot triviality, let alone link equivalence. A geometric invariant of a knot is its genus. This is defined as follows: Seifert (1935) showed how to construct for any knot an orientable surface with the knot as its only edge. There are in fact many such surfaces, but of such surfaces, the one having minimum genus (that is, fewest handles) is called a minimal surface for the knot, and its genus is the genus of the knot. Deforming the knot by Reidemeister moves cannot change the genus. As with the knot group, we have a characterization of triviality in terms of genus, namely: (4.8) The only knot having genus zero is the unknot. This result is the basis of the Haken (1%1) algorithm, refined by Hemion [9] which tests whether a given diagram represents the unknot. Starting with any Seifert surface spanning the knot, the surface is successively modified to produce a sequence of such surfaces of decreasing genus. At each decremental stage a test of minimality of genus is applied and this, in conjunction with (4.8) gives an algorithm for knot triviality. This technique has been further developed to construct an algorithm for deciding whether two diagrams represent isotopic knots, thus demonstrating at least that the problems are decidable even if the algorithms do not appear to be practicable. For details see Waldhausen [15]. I t is tempting to suggest that (4.8) could be used as the basis for a polynomial time nondeterministic algorithm for triviality; this would exist if, for any diagram D representing the trivial knot, there was some polyhedral spanning disc of D which has only polynomially many faces. If this were the case, an NP algorithm for KNOT TRIVIALITY would consist of first guessing a spanning disc, and then verifying that the faces fit together to form a disc and that D is the boundary curve. Unfortunately this very attractive idea fails. Snoeyink [16] has constructed for each integer n, a knot which is trivial but which has a representation in 3 - s p c e by a polygon of 4n + 17 line segments and for which any spanning disc has at least 2n faces. Invariants Reducible to Equivalence
Several of the invariants we have discussed are polynomial time reducible to the problem of knot equivalence. In other words, suppose we define P' (respectively NP') to be the class
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of problems K which can be solved in polynomial time by a deterministic (respectively nondeterministic) Turing machine equipped with an oracle to decide knot isotopy. Then it is clear that: (4.9) KNOT TRIVIALITY E P’
(4.10) CHIRALITY E P’
Here the problem CHIRALITY is defined in the obvious way, given a link diagram, does it represent a chiral link? However, I can only show:
Result 4.11: ALTERNATING Proof:
E NP‘
-
Given a link diagram D , on say n crossings, the nondeterministic machine ‘guesses’ a link K which is alternating and uses the oracle to verify that K D. By Theorem (4.3) K cannot have any more than n crossings. An interesting question is: Problem 4.12:
Does ALTERNATING E P’? A related and slightly easier question is:
Problem 4.13: Show that ALTERNATING E (coNP)! It is also easy to see
Result 4.14: CROSSING NUMBER
E
NP’.
Proof: Given a link diagram D on n crossings and an integer t one can verify that D has crossing number at most t by “guessing” a link diagram D‘ on fewer than t crossings and using Z to show D D’..
-
K2
Figure 11: However, I cannot settle: Problem 4.15:
Does CROSSING NUMBER E PI?
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As far as beknottedness is concerned, it is not clear that finding the beknottedness of a link is decidable. An intriguing question concerns knot primality. If K1, K2 are two knots their sum Kl # K2 is obtained by “tying” them together as shown in Figure 11 and then joining their ends to form a closed string A knot K is prime if is it not the unknot and cannot be expressed as the sum of two nontrivial knots. Consider now the question: KNOT PRIMALITY: Instance: Knot diagram D . Question: Does D represent a prime knot?
It is far from clear that this question is decidable, though it is. The difficulty is that there is no (Ipriori bound on the size of the possible components of a composite knot.
Problem 4.17: DWS KNOT
PRIMALITYE EXPZTMEI?
Note that as a problem about signed graphs, the question can be restated as follows. If K = K1 # K2 and the diagrams D l , D 2 of K1, K2 are given, then the diagram of K is just a block sum D 1* D2. Thus given the right representation of K it is easy to check primality. Just check there is no cut vertex. However, for each composite knot there are infinitely many possible diagrams, some of them have this separation, others do not. There appears to be no bound on the size of the smallest diagram at which the vertex separation must become apparent. 5. Link Polynomials
We close with a brief discussion of complexity questions arising from the recent surge of activity in link polynomial theory which has occurred since the discovery of the Jones polynomial [17] as a link invariant in 1984. There have been several recent excellent surveys of link polynomials and their interrelationship (see for example [18]), and therefore we shall concentrate here just on the Kauffman bracket polynomial. This is essentially the Jones polynomial, and is very close to polynomials already well understood in combinatorics. The bracketpolynomial [ L ]of a link L is obtained from any link diagram of L by applying the equations (5.1)
[xl=A[V,l+B”?
(5.2)
[L 01 = d [Ll,
(5.3)
[OI = d,
locally. We use 0 to denote the unknot. [L] is a polynomial in the 3 variables A , B and d, which are assumed to commute. It thus follows easily that [ ] is well defined on unoriented diagrams. As it stands it is not an invariant of isotopy. However, suppose that we consider the second Reidemeister move, applying the bracket rules we obtain the following formula.
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= AB[
U
A ] + A2 [
U
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1+ 82[ 1+ BA [> c]
=(A2+$+ABd)[n]+AB[)c].
But in order for the bracket to be invariant under isotopy, it certainly must satisfy
and this forces the relations AB = 1 , A 2 + B 2 + A B d = 0
Thus if we specialize the bracket by insisting that B = A - ' , d = - ( A 2 + A -2)
we have shown that the new one variable bracket of L, denoted by (L), is invariant under Reidemeister move (11). It is an easy exercise to check that (L) is also invariant under the third Reidemeister move and this means it is an invariant of regular isotopy. It is not an invariant under Reidemeister move (I). However, provided that the link is oriented and then suitably normalized, Kauffman [19] showed that the resulting invariant is in fact the Jones polynomial of the link. More precisely, if we define the writhe w ( L ) of an oriented link to be the sum of the signs at crossings, then we have
+ve crossing
-ve crossing
Figure 12: Theorem 5.4 The Jones polynomial V,(t) of an oriented link L is given by
As far as complexity is concerned, the writhe is easy to calculate and hence computing the Jones polynomial of an oriented link is Turing equivalent to computing the one variable bracket polynomial of an unoriented version of the same link. But in turn,for the case of alternating links this is easily seen to be Turing equivalent to computing the Tutte polynomial of the underlying graph. To see this note the correspondence between the bracket rule and the deletekontract formulation of the Tutte polynomial
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In general, this is a correspondence between links and signed graphs but in the special (presumably easier) case of alternating link diagrams the edges of the corresponding graph can be chosen to be positive and then we are exactly in the situation of having to determine the Tutte polynomial of a plane graph. Using this correspondence, a consequence of the results of Jaeger, Vertigan and Welsh [20] and Vertigan [21] is: Theorem 5.7: (i)
Determining the Jones polynomial VL(t)of an alternating link is #P-hard.
(ii)
Evaluating the Jones polynomial V,Ct ) at a point to is #P-hard unless to is one of the spe*e4xi13). cia1 points {*I, *i,
One surprising feature of this result is that these special points are exactly the points at which knot theorists knew other evaluations of V L . Working on the assumption that #P # P, made even more reasonable by Toda’s Theorem [22] showing that #Pis at least as hard as any problem in the polynomial hierarchy, we believe there can be no further exact evaluation of the Jones polynomial in terms of easily computable functions. Perhaps the most important open question concerning the Jones polynomial or its bracket equivalent is the following: Probiem 5.9: If V d r ) = 1, is K the unknot?
The answer to this problem is probably not; it is easy to produce examples of non-isotopic knots having the same Jones polynomial in the same way as is it relatively easy to produce nonisomorphic graphs having the same Tutte polynomial. However, settling Problem (5.9) may not be easy: a link K with V d t ) = 1 and K nontrivial will need to be nonaltemating and have crossing number at least 14. The number of distinct knots grows exponentially, see [23] and [24], and combined with Theorem (5.7) we see that a computer search for such a knot does not seem practicable. However, if the answer to Problem (5.9)turned out to be yes, then it would mean that testing knot triviality was no harder than the classical enumeration problems which are known to be #P-complete. It would be a major advance.
Acknowledgement I am grateful to L.H. Kauffman, W.B.R. Lickorish, D.W. Sumners and M.B. Thistlethwaite for their very helpful correspondence and discussions about various points raised in this paper.
References [I] 121 [3]
[4] [5]
[6] [71 [8]
P.G. Tait; On knots I, 11, III, Scienh$c Papers, Volume I, Cambridge University Press, London, 273-347 (1898). G. Burde and H. Zieschang; Knots, de Gruyter (1985). L.H. Kauffman; On Knots, Princeton University Press Ann. Math. Stud. (1987). M.R. Garey and D.S. Johnson; Computers and Intractability - A Guide to the Theory ofNP-Completeness, Freeman, San Francisco (1979). K. Reidemeister;Homotopieringeund Linsemaume, Abh. Math. Sem. Hamburg 11,102-109 (1936). T. Yajima and S. Kinoshita; On the graphs of knots, Osaka Muth. J , 155163 (1953) W. Haken; Theory der Nmalflachen, Acta Math., 105,245375 (l%l). H. Schubert; Knoten und Vollringe, Acta Math., 90,131-286 (1%3).
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G. Hemion, On the classification of homeomorphismsof 2 - m d o l d s and the classification of 3-manfolds, Actu Math., 142,123-155 (1979). M.B. Thistlethwaite;Knot tabulations and related topics, Aspects of Topology, I.M. James and E.H. Kronheimer (editors),Cambridge Univ. Press, 1-76 (1985). W.B.R. Lickorish; The unknotting number of a classical knot, Contemp. Math.. 44,117-121 (1985). K. Murasugi; Jones polynomials and classical conjecturesin knot theory, Topology, 26,187-194 (1987). M.B. Thistlethwaite;A spanning tree expansion of the Jones polynomial, Topology, 26,297-309 (1987). C. MCA. Gordon and J. Luecke; Knots are determined by their complements, J. Amer. Math. SOC.,2, 371415 (1989). F. Waldhausen; Recent results on sufficiently large 3-manifolds. Proc. Symp. Pure Math. Amer. Math. SOC., 32.21-38 (1978). J. Snoeyink; A trivial knot whose spanning disks have exponential size, Proc. 6~ ACM Conj Computational Geometry. 13%147 (1990). V.F.R. Jones; A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. SOC., 12, 103-111 (1985). W.B.R. Lickorish; Polynomials for links, Bull. Lond, Math. SOC.,20,558-588 (1988). L.H. Kauffman;State models and the Jones polynomial, Topology. 26,395407 (1987) F. Jaeger, D.L. Vertigan and D.J.A. Welsh; On the computationalcomplexity of the Jones and Tutte polynomials, Math. Proc. Camb. Phil. SOC.. 108.3S53 (1990).
D.L. V d g a n ; On the computational complexityof Tutte, Homily and Kauffman invariants - to appear. S. Toda; On the computationalpower of PPand $P, Proc. 30thFOCSSymp.. 514-519 (1989) C. Emst and D.W. Sumners; The growth of the number of prime knots, Math. Proc. Camb. Phil. SOC., 102,303-315 (1987). D.J.A. Welsh; On the number of knots and links, Colloq. Math. SOC.Janos Bolyai, 59, 1 4 (1991).
THE COMPLEXITY OF KNOTS Dominic J.A. WELSH Merton College, University of Oxford Oxford, ENGLAND
Abstract The paper considers the computational complexity of classifying knots and of determining several well known knot invariants, both with and without an oracle for testing isotopy.
1.
Introduction
The title of this paper is taken from Tait’s seminal work on knots. Although it is unlikely that Tait was thinking of complexity in the sense that it is used here, the underlying problems encountered by Tait are basically the same questions that we shall be considering. Two fundamental problems of knot theory are:
Is a knotted curve really knotted? Are two knotted curves really the same knot? These were clearly the topics in Tait’s mind when he wrote [l] (p.300): “Before taking up the question of the complexity of a knot a word or two must be said about the methods of reducing any given knot to its
simplest form. I have not been able as yet to find any general method of doing this, nor have I even discovered what would probably solve this diflculty, any perfectly general method ofpronouncing at once from its scheme or otherwise whether a knot is reducible or not.” Tait might be amused to know that one hundred years later and despite a massive effort, these problems are still difficult. For example, until 1974 the knot diagrams shown in Figure 1 were wrongly thought to represent different knots.
Figure 1: In this paper we shall discuss various algorithmic and complexity questions arising from knots. Familiarity with concepts from combinatorics will be assumed; the knot theory concepts will be defined, more details may be found in the books of Burde and Zieschang [2] and Kauffman [3]. The complexity terminology follows Garey and Johnson [4].
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The complexity classes P, NP, #P and EXPTIME have their usual meanings and we remind the reader that although it is known that (1.1)
P G N P G P#'c EXPTIME,
the only inclusion which has been proved to be strict is that P i s distinct from EXPZ7ME. The notation A = B implies A is Turing reducible to B. 2.
Isotopy of Links
A link L with c(L) components in the three-sphere S3 is a smooth sub-manifold that consists of c(L) disjoint simple closed curves. A knot is a link with one component. Two links K, L are isotopic if there exists a homotopy h,: 9 +9 ( 0 I t I 1),such that h , = 1, each h, is a homeomorphism and h, ( K ) = L . We restrict attention to tame links and thus we may assume that, for each link considered, the projection x [ L ] of L to R2 is a finite 4-regular plane graph. The link diagram D(L) of L arising from x[L] is obtained by indicating at each crossing which of the two curve segments goes over the other.
The fundamental theorem of Reidemeister [5]states: Theorem 2.1:
Two links K and L are isotopic if and only if any link diagram of K can be transformed into any link diagram of L by a finite sequence of the moves (I), (II), (111) and their inverses.
Figure 2: Reidemeister moves. We will say that two link diagrams are equivalent if the corresponding links are isotopic. We denote this equivalence of link diagrams and isotopy of links by D -D' and K K'.
-
These moves, known as Reidemeister moves are applied locally. In each case, away from the crossings to which the move is being applied, the diagrams remain unchanged. It is important to note that there are other notions of equivalence; for example one could regard K1 and K2 as equivalent if there exists an autohomeomorphism of 9 which maps K1 to K2. This is a weaker notion than isotopy, for example taking K1 and K2 as shown in Figure 3; they are the right hand trefoil knot and its mirror image the left hand trefoil; these are equivalent in the above sense but are not isotopic; there is no sequence of Reidemeister moves which will transform K 1 to K2. In other words they are chiral. Henceforth we shall exclusively use equivalence in the sense of isotopic equivalence, and thus can rely entirely on the calculus of Reidemeister moves to demonstrate it.
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Figure 3: Two non-isotopic knots. A Graphic Interpretation of Link Isotopy
Given any link diagram D we note that if overlunder crossings are ignored it can be regarded as a 4-regular plane graph G(D).Accordingly it is Eulerian and its dual plane graph, consisting of the faces of G ( D )can be 2-coloured. We color the boundary faces black, and if two faces share a crossing we join them by a signed edge according to the convention shown in Figure 4.
+ve crossing
-ve crossing
Figure 4 In this way, given any link diagram D we get a plane signed graph S(D) in which each edge, appropriately signed, corresponds to a crossing in D.
Conversely, given any plane graph G with its edges signed + or -, we can associate with G, in a canonical way, a link diagram D(G) such that S(D(G))= G. The construction is easy; draw the medial graph of G , call it m(G) and this will be the knot diagram where the over/ under nature of the crossings is determined by the sign of the appropriate edge in G. We leave the details to the reader. From this correspondence between signed planar graphs and link diagrams it is straightforward to verify that the Reidemeister moves (I) -(HI) have the following interpretations as graph transformations Add or delete a loop or isthmus of any sign. (I*): (II*(a)): Insert or delete a parallel pair of oppositely signed edges between any pair of vertices, provided this does not contravene the planar representation. (II*(b)): Contract a pair of oppositely signed edges which are in series. Insert a pair of oppositely signed series edges at a vertex as shown in Figure 5.
(HI*):
The signed star triangle transformation (Figure 6).
Example: Consider the trefoil and its mirror image as shown in Figure 3. Their associated signed graphs D , D a r e shown below. How does one show that K1, K2 are not isotopic? Alternatively, how does one prove that there is no sequence of signed graph moves from D to D?
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Figure 5:
Figure 6 This difficulty is at the heart of the problems considered in this paper. It highlights a property of links which received much attention by Tait, who described K as amphicheiral, (now often called achiral) if K is isotopic to its mirror image. Deciding whether a knot is achiral is still difficult.
Figure 7: As another example of equivalence, take any plane graph G and let each edge be +ve, giving the signed graph G+. Now take the dual plane graph G* with each edge negatively signed, giving (@)-. Then it is not difficult to check
(2.2) The links having diagrams D(G+)and D[( @)-I are isotopic.
Moreover, we can show: (2.3)If the plane graph G has n edges then G+ may be transformed to (G*)- by O(n)Reidemeister moves.
This is just a special case of a more general result:
Result 2.4: For any plane graph G if S denotes a signed version of G, and Sdenotes its e a n a r dual G * with the corresponding edges oppositely signed then S can be transformed to S by O(IE{G)I) of the moves I*-HI*.
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Sketch of Proof: Given a link diagram D with n crossings, which represents the link L , then if A denotes any boundary face of D,and the arc PQ represents the common internal boundary of face A as illustrated in Figure 8 then the diagram fi can be transformed to the link diagram D'by a
Figure 8: sequence of O(n)Reidemeister moves corresponding to pulling PRQ through the diagram. It is now easy to check that the graph of D'is in fact the dual graph of D.We call this move the fold. For more on the relation between graphs and knots see [6] 3. The Complexity of Knot Triviality and Isotopy As mentioned earlier, one of the most fundamental algorithmic questions about knots is to decide whether a knot is trivial. In the now standard terminology of complexity questions we formalize this question as follows: KNOT TRIVIALITY:
Instance: A knot diagram D. Question: Is D topologically equivalent to the unknot? There is an involved and difficult algorithm due to Haken [71, see also Schubert [8] and Hemion [9], which shows that this problem is decidable. However, as far as I am aware, the status of this problem in the complexity hierarchy is not known. More precisely, we pose the fundamental problem: Problem3.1: Find a function f : Z + Z such that if D is a knot diagram which is equivalent to the unknot and which has n crossings then there exists a sequence of at mostf(n) Reidemeister moves which demonstrates the equivalence. More succinctly, we are looking for a bound on the length of Reidemeister proofs of equivalence to the unknot. An example of a knot diagram which in order to be shown equivalent to the unknot needs an increase in the number of crossings, is the diagram K shown in Figure 9. However, in a sense this highlights the inadequacy of the Reidemeister moves as a knot calculus since if one allows the fold as a single move, illustrated by the transition from K to k,then it is not so easy to find examples of diagrams with this property.
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Figure 9 This prompts the question:
Problem 3.2 Find a sequence of link diagrams D ieach representing the unknot and such that the number of crossings, nj, of D icannot be reduced to the unknot in fewer than n: Reidemeister moves. In a discussion with L. Kauffman, we decided that the following construction may provide a class of hard examples to unravel. Take a fairly complicated diagram of the unknot, having say m crossings. Now fold it in half, then fold it in half again and continue in this way for perhaps k folds. This will give a diagram D which still represents the unknot, has approximately q m k ) crossings, and would seem to demand very many Reidemeister moves in any unraveling process. However, proving this assertion seems very difficult. A closely related, but even harder problem, is the following: KNOT EQUIVALENCE:
Instance: Two knot diagrams D1, D2. Question: Do Dland 4 represent equivalent knots? It would be a major advance to show that KNOT EQUIVALENCE belonged to the complexity class EXPTZME; though again there is no firm evidence indicating that the problem is not in polynomial time P.
4. Classical Link Invariants Two link diagrams which have different numbers of components obviously represent non isotopic links. This is probably the simplest example of a link invariant and is used as a first subdivision of links in the classical tables of links (see for example [lo]). A less trivial invariant is the crossing number. This is defined to be the minimum number of crossings in any diagram representing the link. Determining the crossing number of a link is difficult. Expressed more formally, define the problem: CROSSING NUMBER:
Instance: A knot K represented by a diagram D , together with an integer k. Question: Is the crossing number of K < k? Since a knot has zero crossing number if and only if it is trivial, we have: (4.1) KNOT TRIVIALITY = CROSSING NUMBER.
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Tait introduced another knot invariant which is superficially related to crossing number and which he called beknottedness. It is the minimum number of changes of sign of crossings which reduce the knot to triviality. He remarks [l] (p.308) “There must be some very simple method of determining the amount of beknottedness for any given knot: but I have not hit upon it. ” He might have been surprised to know that determining the beknottedness of a knot is still an immensely difficult problem, even for diagrams with as few as 9 crossings, see for example [Ill. A link diagram D is alternating if and only if it has the property that in D the crossings are alternately overlunderlover.. .. While it is trivial to verify that a given diagram is alternating, it is highly nontrivial to test whether a given link L has a representation as an alternating link diagram. Any such link is called alternating. Equivalently, a link diagram D is alternating if and only if the associated signed graph has all its edges the same sign. This is easy to see. Thus, the question of deciding whether a link is alternating reduces to: Problem 4 3:
Is there a polynomial time algorithm which will check whether a given signed graph is transformable to a monosigned graph using the moves I*-HI*? If there were such an algorithm it would settle the knot triviality question, since a recent result of Murasugi [12] and Thistlethwaite [13], which settles one of the long standing Tait conjectures, states:
Theorem 43: If L is an alternating link and D is an alternating diagram representing L then provided D has no nugatory crossing it has a minimum number of crossings over all diagrams representing L. (A crossing is nugatory in D if, as is shown in Figure 10, it corresponds exactly to the corresponding edge of the graph G(D)being an isthmus).
Figure 10: A nugatory crossing. It is clear from this and the fact that nugatory crossings are easily recognized and eliminated, that finding an alternating representation of an alternating knot is at least as hard (in the computational sense) as deciding whether a knot is trivial. Hence we define the problem: ALTERNATING: Instance: A link diagram D. Question: Does D represent an alternating link?
We have immediately (4.4) KNOT TRIVIALITY = ALTERNATING.
One of the most powerful invariants of a knot is its group, or more precisely the funda-
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mental group of the knot complement, and it is natural to examine its complexity. The origin of this invariant can be traced back to Poincark. A series of results culminating in a recent
result of Gordon and Luecke [14] show that it is close to providing a complete classification of knots. First we observe:
(4.5)From a knot diagram it is easy to obtain in polynomial time a presentation of the knot group in terms of generators and relations. The well known Wirtinger presentation of an ncrossing diagram gives a presentation of n generators and n relations and can be found in time o(n2). (4.6) The famous theorem of Dehn and Papkyriakopoulos (1957) states that K has a group presented by a single generator if and only if K is the unknot. In other words:
(4.7) The trivial knot is completely characterized by its group. Unfortunately this characterization is not always easy to use and at the moment it still does not seem possible to use it to produce even an exponential time algorithm for deciding knot triviality, let alone link equivalence. A geometric invariant of a knot is its genus. This is defined as follows: Seifert (1935) showed how to construct for any knot an orientable surface with the knot as its only edge. There are in fact many such surfaces, but of such surfaces, the one having minimum genus (that is, fewest handles) is called a minimal surface for the knot, and its genus is the genus of the knot. Deforming the knot by Reidemeister moves cannot change the genus. As with the knot group, we have a characterization of triviality in terms of genus, namely: (4.8) The only knot having genus zero is the unknot. This result is the basis of the Haken (1%1) algorithm, refined by Hemion [9] which tests whether a given diagram represents the unknot. Starting with any Seifert surface spanning the knot, the surface is successively modified to produce a sequence of such surfaces of decreasing genus. At each decremental stage a test of minimality of genus is applied and this, in conjunction with (4.8) gives an algorithm for knot triviality. This technique has been further developed to construct an algorithm for deciding whether two diagrams represent isotopic knots, thus demonstrating at least that the problems are decidable even if the algorithms do not appear to be practicable. For details see Waldhausen [15]. I t is tempting to suggest that (4.8) could be used as the basis for a polynomial time nondeterministic algorithm for triviality; this would exist if, for any diagram D representing the trivial knot, there was some polyhedral spanning disc of D which has only polynomially many faces. If this were the case, an NP algorithm for KNOT TRIVIALITY would consist of first guessing a spanning disc, and then verifying that the faces fit together to form a disc and that D is the boundary curve. Unfortunately this very attractive idea fails. Snoeyink [16] has constructed for each integer n, a knot which is trivial but which has a representation in 3 - s p c e by a polygon of 4n + 17 line segments and for which any spanning disc has at least 2n faces. Invariants Reducible to Equivalence
Several of the invariants we have discussed are polynomial time reducible to the problem of knot equivalence. In other words, suppose we define P' (respectively NP') to be the class
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of problems K which can be solved in polynomial time by a deterministic (respectively nondeterministic) Turing machine equipped with an oracle to decide knot isotopy. Then it is clear that: (4.9) KNOT TRIVIALITY E P’
(4.10) CHIRALITY E P’
Here the problem CHIRALITY is defined in the obvious way, given a link diagram, does it represent a chiral link? However, I can only show:
Result 4.11: ALTERNATING Proof:
E NP‘
-
Given a link diagram D , on say n crossings, the nondeterministic machine ‘guesses’ a link K which is alternating and uses the oracle to verify that K D. By Theorem (4.3) K cannot have any more than n crossings. An interesting question is: Problem 4.12:
Does ALTERNATING E P’? A related and slightly easier question is:
Problem 4.13: Show that ALTERNATING E (coNP)! It is also easy to see
Result 4.14: CROSSING NUMBER
E
NP’.
Proof: Given a link diagram D on n crossings and an integer t one can verify that D has crossing number at most t by “guessing” a link diagram D‘ on fewer than t crossings and using Z to show D D’..
-
K2
Figure 11: However, I cannot settle: Problem 4.15:
Does CROSSING NUMBER E PI?
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As far as beknottedness is concerned, it is not clear that finding the beknottedness of a link is decidable. An intriguing question concerns knot primality. If K1, K2 are two knots their sum Kl # K2 is obtained by “tying” them together as shown in Figure 11 and then joining their ends to form a closed string A knot K is prime if is it not the unknot and cannot be expressed as the sum of two nontrivial knots. Consider now the question: KNOT PRIMALITY: Instance: Knot diagram D . Question: Does D represent a prime knot?
It is far from clear that this question is decidable, though it is. The difficulty is that there is no (Ipriori bound on the size of the possible components of a composite knot.
Problem 4.17: DWS KNOT
PRIMALITYE EXPZTMEI?
Note that as a problem about signed graphs, the question can be restated as follows. If K = K1 # K2 and the diagrams D l , D 2 of K1, K2 are given, then the diagram of K is just a block sum D 1* D2. Thus given the right representation of K it is easy to check primality. Just check there is no cut vertex. However, for each composite knot there are infinitely many possible diagrams, some of them have this separation, others do not. There appears to be no bound on the size of the smallest diagram at which the vertex separation must become apparent. 5. Link Polynomials
We close with a brief discussion of complexity questions arising from the recent surge of activity in link polynomial theory which has occurred since the discovery of the Jones polynomial [17] as a link invariant in 1984. There have been several recent excellent surveys of link polynomials and their interrelationship (see for example [18]), and therefore we shall concentrate here just on the Kauffman bracket polynomial. This is essentially the Jones polynomial, and is very close to polynomials already well understood in combinatorics. The bracketpolynomial [ L ]of a link L is obtained from any link diagram of L by applying the equations (5.1)
[xl=A[V,l+B”?
(5.2)
[L 01 = d [Ll,
(5.3)
[OI = d,
locally. We use 0 to denote the unknot. [L] is a polynomial in the 3 variables A , B and d, which are assumed to commute. It thus follows easily that [ ] is well defined on unoriented diagrams. As it stands it is not an invariant of isotopy. However, suppose that we consider the second Reidemeister move, applying the bracket rules we obtain the following formula.
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U
A ] + A2 [
U
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1+ 82[ 1+ BA [> c]
=(A2+$+ABd)[n]+AB[)c].
But in order for the bracket to be invariant under isotopy, it certainly must satisfy
and this forces the relations AB = 1 , A 2 + B 2 + A B d = 0
Thus if we specialize the bracket by insisting that B = A - ' , d = - ( A 2 + A -2)
we have shown that the new one variable bracket of L, denoted by (L), is invariant under Reidemeister move (11). It is an easy exercise to check that (L) is also invariant under the third Reidemeister move and this means it is an invariant of regular isotopy. It is not an invariant under Reidemeister move (I). However, provided that the link is oriented and then suitably normalized, Kauffman [19] showed that the resulting invariant is in fact the Jones polynomial of the link. More precisely, if we define the writhe w ( L ) of an oriented link to be the sum of the signs at crossings, then we have
+ve crossing
-ve crossing
Figure 12: Theorem 5.4 The Jones polynomial V,(t) of an oriented link L is given by
As far as complexity is concerned, the writhe is easy to calculate and hence computing the Jones polynomial of an oriented link is Turing equivalent to computing the one variable bracket polynomial of an unoriented version of the same link. But in turn,for the case of alternating links this is easily seen to be Turing equivalent to computing the Tutte polynomial of the underlying graph. To see this note the correspondence between the bracket rule and the deletekontract formulation of the Tutte polynomial
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In general, this is a correspondence between links and signed graphs but in the special (presumably easier) case of alternating link diagrams the edges of the corresponding graph can be chosen to be positive and then we are exactly in the situation of having to determine the Tutte polynomial of a plane graph. Using this correspondence, a consequence of the results of Jaeger, Vertigan and Welsh [20] and Vertigan [21] is: Theorem 5.7: (i)
Determining the Jones polynomial VL(t)of an alternating link is #P-hard.
(ii)
Evaluating the Jones polynomial V,Ct ) at a point to is #P-hard unless to is one of the spe*e4xi13). cia1 points {*I, *i,
One surprising feature of this result is that these special points are exactly the points at which knot theorists knew other evaluations of V L . Working on the assumption that #P # P, made even more reasonable by Toda’s Theorem [22] showing that #Pis at least as hard as any problem in the polynomial hierarchy, we believe there can be no further exact evaluation of the Jones polynomial in terms of easily computable functions. Perhaps the most important open question concerning the Jones polynomial or its bracket equivalent is the following: Probiem 5.9: If V d r ) = 1, is K the unknot?
The answer to this problem is probably not; it is easy to produce examples of non-isotopic knots having the same Jones polynomial in the same way as is it relatively easy to produce nonisomorphic graphs having the same Tutte polynomial. However, settling Problem (5.9) may not be easy: a link K with V d t ) = 1 and K nontrivial will need to be nonaltemating and have crossing number at least 14. The number of distinct knots grows exponentially, see [23] and [24], and combined with Theorem (5.7) we see that a computer search for such a knot does not seem practicable. However, if the answer to Problem (5.9)turned out to be yes, then it would mean that testing knot triviality was no harder than the classical enumeration problems which are known to be #P-complete. It would be a major advance.
Acknowledgement I am grateful to L.H. Kauffman, W.B.R. Lickorish, D.W. Sumners and M.B. Thistlethwaite for their very helpful correspondence and discussions about various points raised in this paper.
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