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The rank of a warping matrix
Topology and its Applications 206 (2016) 228–240
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Topology and its Applications www.elsevier.com/locate/topol
The rank of a warping matrix Taira Akiyama a , Ayaka Shimizu b,∗ , Ryohei Watanabe a a
Department of Information and Computer Engineering, Gunma National College of Technology, 580 Toriba-cho, Maebashi-shi, Gunma, 371-8530, Japan b Department of Mathematics, Gunma National College of Technology, 580 Toriba-cho, Maebashi-shi, Gunma, 371-8530, Japan
a r t i c l e
i n f o
Article history: Received 25 August 2015 Received in revised form 15 March 2016 Accepted 5 April 2016 Available online 12 April 2016 MSC: 57M25
a b s t r a c t The warping matrix has been defined for knot projections and knot diagrams by using warping degrees, and the warping matrix of a knot diagram represents the knot diagram uniquely. In this paper we show that the rank of a warping matrix is one greater than the crossing number of the knot projection or diagram. We also discuss the linearly independence of knot diagrams by considering the warping incidence matrices. © 2016 Elsevier B.V. All rights reserved.
Keywords: Knot diagram Warping degree Warping matrix
1. Introduction In [8], Kawauchi introduced the notion of warping degrees of oriented knot and link diagrams. In [14] and [15], the properties of the warping degrees of knot and link diagrams were studied. For example, inequalities of the warping degree and the crossing number of a knot and link diagram characterizing alternating diagrams were given there. Warping degrees of spatial graphs, virtual knots and nanowords are also studied (see, for example, [4,7,9,11]). In [17] and [10], the warping polynomial and the warping crossing polynomial of a knot diagram were defined via the warping degree labeling and the warping degree sequence which have the information of warping degrees of the diagram, and studied (see also [5]) and applied to define polynomial invariants for virtual knots (see [1,2,6]). The span of a warping polynomial and a warping degree sequence was also studied in [17] and applied to define an alternating distance in [13]. (Recently, it was shown in [12] that every knot has a diagram such that the span is two.) In [16], the warping matrix of a * Corresponding author. E-mail address: [email protected] (A. Shimizu). http://dx.doi.org/10.1016/j.topol.2016.04.003 0166-8641/© 2016 Elsevier B.V. All rights reserved.
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Fig. 1. d(Da ) = d(Da ) + 1 and d(Db ) = d(Db ) − 1.
knot projection and a knot diagram was defined by the second author, which has the information of warping degrees and warping polynomials, and it is shown that there is a one-to-one correspondence between oriented knot diagrams on S 2 and warping matrices. Hence the warping matrix can be used as a notation for oriented knots. However, it is not easy to calculate warping matrices for knot projections and diagrams with a large crossing number; the size of the warping matrix of a knot projection with c crossings is 2c × 2c, and that of a knot diagram is (2c − 1) × 2c. One of the motivations on the study of warping matrices is to reduce the size of the matrix. The following theorem would be helpful if we can approach this matter with the theory of matrix decompositions such as the LDU decomposition: Theorem 1.1. Let P be an oriented knot projection on S 2 , and M (P ) the warping matrix of P . We have the following equality: rank M (P ) = c(P ) + 1, where c(P ) is the crossing number of P . Let D be an oriented knot diagram on S 2 , and c(D) the crossing number of D. Let M (D) be the warping matrix of D without signs, which is mentioned concretely in Section 2. We also have the following theorem: Theorem 1.2. We have rank M (D) = c(D) + 1. The rest of this paper is organized as follows: In Section 2, we review the warping matrix. In Section 3, we prove Theorems 1.1 and 1.2. In Section 4, we discuss warping matrices without signs for the trivial knots and prime alternating knots. In Section 5, we define a new matrix, the warping incidence matrix of a knot diagram. In Section 6, we investigate the linearly independence for knot diagrams. 2. Warping matrix In this section we review the warping degree and warping matrix. See [14] and [16] for details. In this paper, we assume that all knot projections and diagrams are oriented. Let D be a knot diagram on S 2 . Take a base point of D avoiding crossing points. We denote by Db the pair of D and b. A crossing point p is said to be a warping crossing point of Db if we meet p as an undercrossing first (and an overcrossing later) when we travel D from b (we encounter each crossing twice). The warping degree d(Db ), which was defined by Kawauchi, of Db is defined to be the number of the warping crossing points of Db . As shown in [14], we have the following: Proposition 2.1. (Lemma 2.5 in [14]) When a base point is moved through an overcrossing (resp. undercrossing) on a knot diagram with the orientation, the warping degree increases (resp. decreases) by one. That is, we have d(Da ) = d(Da ) + 1 and d(Db ) = d(Db ) − 1 for a knot diagram D with the base points illustrated in Fig. 1.
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Fig. 2. Warping matrix.
A knot projection or diagram is said to be trivial if it has no crossings. Let P be a nontrivial knot projection on S 2 with c crossing points. Take a base point at each edge of P , where an edge means a part of P between two crossings which has no crossings in the interior. Label them b1, b2 , . . . , b2c from an edge following the orientation of the diagram. From P , we obtain 2c knot diagrams by giving over/under c information at each crossing of P . We call them D1 , D2 , . . . , D2 . The warping matrix M (P ) of P is the 2c × 2c matrix defined by: M (P ) = ( aij ), where aij = d(Dbi j ). An example is shown in Fig. 2. We consider warping matrices up to permutations on rows and cyclic permutations on columns. Each row of M (P ) is said to be the warping degree sequence of the corresponding diagram. As mentioned in Proposition 2.2 in [16], warping matrices of knot projections have the following properties: • On each row, the difference of two elements which are next to each other is one. c • On each column, n appears times (n = 0, 1, 2, . . . , c). n The ou matrix U (P ) of P is the matrix obtained from M (P ) by the multiplication U (P ) = M (P ) × A, where A is the 2c × 2c matrix as follows: ⎛
−1 ⎜ 1 ⎜ ⎜ ⎜ 0 A=⎜ ⎜ .. ⎜ . ⎜ ⎝ 0 0 We give an example.
0 −1 1 .. . 0 0
... ... ... .. . ... ...
0 0 0 .. . −1 1
⎞ 1 0 ⎟ ⎟ ⎟ 0 ⎟ .. ⎟ ⎟. . ⎟ ⎟ 0 ⎠ −1
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Fig. 3. Knot projection P .
Example 2.2. The warping matrix M (P ) of the knot projection P depicted in Fig. 3 is ⎛
0 ⎜1 ⎜ M (P ) = ⎜ ⎝1 2
1 2 0 1
2 1 1 0
⎞ 1 2⎟ ⎟ ⎟ 0⎠ 1
and the ou matrix of P is: ⎛
01 ⎜1 2 ⎜ U (P ) = ⎜ ⎝1 0 21 ⎛
⎞⎛ −1 1 ⎜ ⎟ 2⎟⎜ 1 ⎟⎜ 0⎠⎝ 0 0 1
2 1 1 0
1 ⎜ 1 ⎜ =⎜ ⎝ −1 −1
1 −1 1 −1
−1 1 −1 1
0 0 −1 0 1 −1 0 1 ⎞ −1 −1 ⎟ ⎟ ⎟. 1 ⎠ 1
⎞ 1 0 ⎟ ⎟ ⎟ 0 ⎠ −1
Each row of U (P ) corresponds to a knot diagram, each column corresponds to a crossing, and each element represents the over/under information where 1 means over and −1 means under. Since we pass each crossing twice, there are just c pairs of columns uniquely such that the sum of them is 0. For example, U (P ) of Example 2.2 has the pairs 1st and 4th, and 2nd and 3rd. From the pairing, we can recover the Gauss diagram of P . For details, see [16]. We define the warping matrix without signs for knot diagrams. Let D be an oriented knot diagram on S 2 , and P the knot projection obtained from D by ignoring the over/under information. We define the warping matrix M (D) to be the matrix acquired from M (P ) by deleting the row of D, which is the warping degree sequence of D. We can also obtain M (D) from the ordinary warping matrix M (D) just by removing the signs of elements. For example, we have ⎛
M(
0 ⎜1 ⎜1 ⎜ )=⎜ ⎜2 ⎜2 ⎝ 2 3
1 0 2 1 1 3 2
2 1 3 0 2 2 1
3 2 2 1 1 1 0
2 3 1 2 2 0 1
⎞ 1 2⎟ 0⎟ ⎟ 3⎟ ⎟ 1⎟ ⎠ 1 2
(cf. Fig. 2). 3. Proof of Theorems 1.1 and 1.2 In this section we prove Theorems 1.1 and 1.2. We first show the following lemma: Lemma 3.1. Let c be an integer which is greater than 1. Let a1 , a2 , . . . , ac , ac+1 be integers. We have the following equation.
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a1
a
2
.
..
ac−1
ac
ac+1
−1 1 .. . 1 1 −1
1 −1 .. . 1 1 −1
... ... .. . ... ... ...
1 1 .. . −1 1 −1
c
= −2c−1 ai + (c − 2)ac+1 .
1
i=1
−1
−1
1 1
(1)
Proof. By adding the (c + 1)th row to the other rows, the equation (1) is equivalent to:
a1 + ac+1
a +a c+1
2
..
.
ac−1 + ac+1
ac + ac+1
ac+1
−2 0 .. . 0 0 −1
0 −2 .. . 0 0 −1
... ... .. . ... ... ...
0 0 .. . −2 0 −1
c
= −2c−1 ai + (c − 2)ac+1 .
0
i=1
−2
−1
0 0
We will show the equation (2) by an induction on c. • For c = 2, we have
a + a −2 0
3
1
a2 + a3 0 −2 = −2(a1 + a2 ).
a3 −1 −1
Hence (2) holds for c = 2. • Assume (2) holds for c = k − 1 for an integer k greater than two. Now we consider the case c = k:
a1 + ak+1
a +a k+1
2
..
.
ak−1 + ak+1
ak + ak+1
ak+1
−2 0 .. . 0 0 −1
0 −2 .. . 0 0 −1
... ... .. . ... ... ...
0 0 .. . −2 0 −1
.
0
−2
−1
0 0
The Laplace expansion along the (k + 1)th column yields:
a1 + ak+1 −2 0 . . . 0
a +a 0 −2 . . . 0
k+1
2
.. .. .. . .
(−1)k+(k+1) × (−2)
. . . .
ak−1 + ak+1 0 0 . . . −2
−1 −1 . . . −1
ak+1
a1 + ak+1 −2 0 . . . 0
a +a 0 −2 . . . 0
k+1
2
.. .. .. . .
. + (−1)(k+1)+(k+1) × (−1)
. . . .
ak−1 + ak+1 0 0 . . . −2
ak + ak+1 0 0 . . . 0
Laplace expansion again along the kth row at the second term yields:
(2)
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a1 + ak+1
a +a k+1
2
.
.. 2
ak−1 + ak+1
ak+1
−2 0 .. . 0 −1
0 −2 .. . 0 −1
... ... .. . ... ...
−2
0
− (−1)k+1 (ak + ak+1 ) .
.
.
−2
0 −1
0 0
0 −2 .. . 0
... ... .. . ...
0
0
..
. .
−2
233
(3)
We remark that the (k, 1) element of the first matrix in (3) is ak+1 . By assumption, (3) is equal to 2 −2
(k−1)−1
= −2
k−1
k−1
k−1
ai + (k − 1 − 2)ak+1
i=1
ai + (k − 3)ak+1
− 2k−1 (ak + ak+1 )
i=1
= −2
k−1
k−1
= −2
k−1
ai + (k − 3)ak+1 + ak + ak+1
i=1 k
− (−1)k+1 (ak + ak+1 )(−2)k−1
ai + (k − 2)ak+1
.
i=1
Hence (2) holds. 2 By multiplying by −1 the (c + 1)th column, we obtain the following corollary from Lemma 3.1: Corollary 3.2. We have
a1
a
2
.
..
ac−1
ac
ac+1
−1 1 .. . 1 1 −1
1 −1 .. . 1 1 −1
... ... .. . ... ... ...
1 1 .. . −1 1 −1
−1
−1
c
c−1
=2 ai + (c − 2)ac+1 .
−1
i=1
1
1
(4)
Now we prove Theorem 1.1. Proof of Theorem 1.1. For a 2c × 2c matrix M (P ) = (a1 , a2 , . . . , a2c ), subtract a2c−1 from a2c , a2c−2 from a2c−1 , a2c−3 from a2c−2 , . . . , and a1 from a2 . Let (a, v 1 , v 2 , . . . , v 2c−1 ) be the matrix obtained by the procedure. Note that (v 1 , v 2 , . . . , v 2c−1 ) is a submatrix of U (P ). By the property of ou matrix, there are just (c − 1) pairs of columns v i and v j uniquely such that v i + v j = 0. For each pair, add one to the other. By reordering some columns v i , we have (a, v ϕ1 , v ϕ2 , . . . , v ϕc , 0, 0, . . . , 0). Hence rank M (P ) ≤ c(P ) + 1. Next we show that a, v ϕ1 , v ϕ2 , . . . , v ϕc are linearly independent. Since the columns v ϕ1 , v ϕ2 , . . . , v ϕc correspond to all the c crossings, and the rows correspond to all the over/under information, we can obtain
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the following (c + 1) × (c + 1) submatrix of (a, v ϕ1 , v ϕ2 , . . . , v ϕc ) by reordering some rows ⎛
−1 1 .. . 1 1 −1
a1 ⎜ a ⎜ 2 ⎜ . ⎜ .. ⎜ ⎜ ⎜ ac−1 ⎜ ⎝ ac ac+1
1 −1 .. . 1 1 −1
... ... .. . ... ... ...
1 1 .. . −1 1 −1
⎞ 1 1 ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ 1 ⎟ ⎟ −1 ⎠ −1
where ai is an element of a (i = 1, 2, . . . , c + 1). By Lemma 3.1, the determinant of the submatrix is −2
c
c−1
ai + (c − 2)ac+1
.
i=1
Remark that the elements of a are all non-negative and there are at most one 0. Hence the determinant is non-zero. Therefore a, v ϕ1 , v ϕ2 , . . . , v ϕc are linearly independent. Hence rank M (P ) = c(P ) + 1.
2
We prove Theorem 1.2. Proof of Theorem 1.2. Similar to Theorem 1.1 except that M (D) has one row fewer than M (P ). Remark that we can obtain at least one of the following two submatrices ⎛
a1 ⎜ a ⎜ 2 ⎜ . ⎜ .. ⎜ ⎜ ⎜ ac−1 ⎜ ⎝ ac ac+1
−1 1 .. . 1 1 −1
1 −1 .. . 1 1 −1
... ... .. . ... ... ...
1 1 .. . −1 1 −1
⎛ ⎞ a1 1 ⎜ ⎟ 1 ⎟ ⎜ a2 ⎜ . ⎟ ⎜ . ⎟ ⎟ and ⎜ . ⎜ ⎟ ⎜ ac−1 1 ⎟ ⎜ ⎟ ⎝ ac −1 ⎠ −1 ac+1
which are the matrices of Lemma 3.1 and Corollary 3.2.
−1 1 .. . 1 1 −1
1 −1 .. . 1 1 −1
... ... .. . ... ... ...
1 1 .. . −1 1 −1
⎞ −1 −1 ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ −1 ⎟ ⎟ 1 ⎠ 1
2
4. Warping matrices for the trivial knot and prime alternating knots In this section we investigate warping degrees and warping matrices and discuss the warping matrices for the trivial knot and prime alternating knots. We first show the following lemma: Lemma 4.1. Let Nc = 0
c c c c +1 +2 + ··· + c 0 1 2 c
for a positive integer c. Then we have Nc = 2c−1 c. Proof. From the binomial theorem, we have c c c c 2 = + + + ··· + . 0 1 2 c c
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When c is odd, we have 2c = 2
c c c c , + + + · · · + c−1 0 1 2 2
and then we have c c c c + + + · · · + c−1 = 2c−1 . 0 1 2 2 Similarly, when c is even, we have c c c c c + c , 2 =2 + + + ··· + c − 1 0 1 2 2 2 c
and then we have c c c c 1 c + + + ... c = 2c−1 − . 0 1 2 2 2c 2 −1 Hence we have c c c c Nc = c + + + · · · + c−1 0 1 2 c = c × 2c−1 for an odd integer c, and c c c c c c + Nc = c + + + ··· + c 2 2c 0 1 2 2 −1 c c 1 c = c 2c−1 − + = 2c−1 c 2 2c 2 2c for an even integer c. 2 We have the following proposition: Proposition 4.2. Let M (P ) be the warping matrix of a knot projection P with c crossings. We have (1 1 . . . 1)M (P ) = (2c−1 c 2c−1 c . . . 2c−1 c), where (1 1 . . . 1) is the row vector with length 2c whose elements are all 1, and (2c−1 c 2c−1 c . . . 2c−1 c) is the row vector with length 2c whose elements are all 2c−1 c. Proof. The row vector (1 1 . . . 1)M (P ) represents the sum of all the rows of M (P ). As we have seen in Section 2 (and [10,16]), n appears nc times at each column of M (P ) (n = 0, 1, 2, . . . , c). Hence each element of (1 1 . . . 1)M (P ) is c c c c +1 +2 + ··· + c 0 1 2 c
0
and this is equal to 2c−1 c by Lemma 4.1.
2
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Fig. 4. The knot diagram D of the trivial knot has the warping degree sequence 2123432.
Fig. 5. A connected sum P Q of P and Q.
Fig. 6. A Gauss diagram represents a connected sum if and only if the chords can be divided into two sides, where the connecting edges separate the chords into two sides.
Since (1 1 . . . 1)M (D) also represents the sum of all the rows of M (D), we can reobtain the warping degree sequence of D with c crossings from M (D) by (2c−1 c 2c−1 c . . . 2c−1 c) − (1 1 . . . 1)M (D). By the definition of the warping degree, we can say that a knot diagram D represents the trivial knot if D has zero in the warping degree sequence. Note that the converse does not hold (see, for example, Fig. 4). In terms of matrices, we have the following proposition: Proposition 4.3. Let M (D) be a 2c−1 × 2c warping matrix without signs. If (2c−1 c 2c−1 c . . . 2c−1 c) − (1 1 . . . 1)M (D) has zero as an element, then M (D) represents the trivial knot. Next we discuss about a connected sum on knot projections and diagrams. For two knot projections (resp. diagrams) P and Q, we say that the knot projection (resp. diagram) depicted in the right hand side in Fig. 5 is a connected sum of P and Q, and denote it by P Q. We say a knot projection (resp. diagram) is prime if it is not any connected sum of two nontrivial knot projections (resp. diagrams). In this paper, we call the two edges of P Q connecting P and Q the connecting edges of P and Q. It is known that a Gauss diagram represents a connected sum if and only if the chords of the Gauss diagram can be divided into two sides as shown in Fig. 6 (see, for example, [3]). Connecting edges have the same warping degrees as follows: Proposition 4.4. Let D = P Q be a knot diagram which is a connected sum of P and Q. Let e1 , e2 be the connecting edges of P and Q and let b1 , b2 be a base point on e1 , e2 , respectively. Then we have d(Db1 ) = d(Db2 ). Proof. From e1 to e2 , we go through every crossing of just P (or Q) twice; as an overcrossing once and as an undercrossing once. Hence Db1 and Db2 have the same warping degree because of the property of the warping degree in Proposition 2.1. 2
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Fig. 7. A chord has e and f at the opposite sides.
Moreover, we have the following proposition: Proposition 4.5. A warping matrix M (P ) is a warping matrix of a knot projection which is a connected sum of two nontrivial knot projections if and only if there exists a pair of the same columns in M (P ). Proof. If a knot projection P is a connected sum, the connecting edges have the same warping degree with any over/under information at the crossings of P by Proposition 4.4. Hence the corresponding columns in M (P ) have all the same elements. Next we will show that if M (P ) has a pair of the same columns then the corresponding edges are connecting edges by showing the contraposition. Let P be a knot projection and let e, f be a pair of edges of P which is not any pair of connecting edges. Since the chords of the Gauss diagram of P cannot be separated by e and f , there exists a chord which has e and f at the opposite sides in the Gauss diagram (see Fig. 7). We call v the corresponding crossing of P . Now we will show that we can give over/under information to the crossings of P so that e and f have different warping degrees when we take base points there. Let D be a knot diagram obtained from P by giving over/under information at each crossing. If e and f have the same warping degrees, apply a crossing change at the crossing v. Then, as shown in Theorem 1 (ii) in [10], the warping degrees with base points on the edges from the overcrossing of v to the undercrossing of v decreases by one, and the warping degrees with base points on the edges from the undercrossing to overcrossing increases by one. Then the warping degrees of e and f differ by the crossing change. Hence the columns corresponding to e and f have different elements in M (P ). 2 For M (D), we have the following: Corollary 4.6. A warping matrix without signs M (D) is a warping matrix without signs of a knot diagram which is a connected sum of two nontrivial knot diagrams if and only if there exists a pair of the same columns in M (D). As mentioned in [17], the warping degree sequence of an alternating knot is also alternating like (k k + 1 k . . . k + 1). It is also shown in [17] that a knot diagram D is alternating if and only if the span of the warping degree sequence is one, where the span of a warping degree sequence is the difference of the maximal entry and the minimal entry in the sequence. We have the following corollary: Corollary 4.7. A 2c−1 × 2c warping matrix without signs M (D) represents a minimal crossing diagram of a prime alternating knot with the crossing number c (= rank M (D) − 1) if and only if • M (D) does not have any pair of the same columns, and • the span of (2c−1 c 2c−1 c . . . 2c−1 c) − (1 1 . . . 1)M (D) is one. 5. Warping incidence matrix In this section we define the warping incidence matrix for knot diagrams. Let D be a knot diagram on S 2 with c crossings. Label the edges e1 , e2 , . . . , e2c from an edge following the orientation of the diagram. Label the crossings v1 , v2 , . . . , vc . The warping incidence matrix m(D) of D is defined as follows:
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Fig. 8. Warping incidence matrix.
Fig. 9. A crossing and four edges.
m(D) = (aij ),
where aij =
1 (vi is a warping crossing point of Dbj ); 0 (vi is not a warping crossing point of Dbj ),
where bj is a base point on ej . An example is shown in Fig. 8. We consider warping incidence matrices up to permutations on rows and cyclic permutations on columns. By definition, we have the following: Proposition 5.1. The sum of all the rows of the warping incidence matrix of a knot diagram D is the warping degree sequence of D. We show the following proposition. Proposition 5.2. Let m(D) = (aij ) be the warping incidence matrix of a knot diagram D with c(D) = c. For any integer k ∈ {1, 2, . . . , c}, there exists a unique integer l ∈ {1, 2, . . . , 2c} such that ⎧ ⎪ ⎨ ak l−1 = 0, ak l = 1 and ⎪ ⎩a i l−1 = ai l (i = k).
(5)
Proof. See Fig. 9. At the kth row in m(D), 1 appears from lth column through mth column because the corresponding edges have the crossing vk as a warping crossing point, whereas 0 appears from (m + 1)th column through (l − 1)th column. Hence we have ak l−1 = 0 and ak l = 1. Since the two edges el−1 and el have the same warping crossing points except at vk , we have ai l−1 = ai l for i = k. 2 We have the following corollaries: Corollary 5.3. Let D be a knot diagram and let Di be the diagram obtained from D by a crossing change at the crossing vi . Then m(Di ) is obtained from m(D) by switching 0 and 1 at the row corresponding to vi . Corollary 5.4. We can obtain the Gauss diagram of D without signs from m(D). We also define the signed warping incidence matrix of a knot diagram to be the matrix obtained from the warping incidence matrix by replacing “1” with “−1” or “1” at each row corresponding to a negative crossing. Then we have the following corollary:
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Corollary 5.5. We can obtain the Gauss diagram and the knot diagram on S 2 from a signed warping incidence matrix. Next we show the following lemma: Lemma 5.6. Let D be a knot diagram with c(D) = c. Let m(D) be the warping incidence matrix of D, and v i the ith row of m(D) (i = 1, 2, . . . , c). Then v 1 , v 2 , . . . , v c and 1 are linearly independent, where 1 is the row vector (11 . . . 1) with length 2c. Proof. Let λ1 v 1 + λ2 v 2 + · · · + λc v c + λc+1 1 = 0.
(6)
For each k ∈ {1, 2, . . . , c}, there exists an integer l satisfying (5) in Proposition 5.2. Since the lth and (l−1)th elements of (6) are zero, the difference of them, which is λk , is also zero. Hence we have λ1 = λ2 = · · · = λc = 0 and therefore λc+1 = 0. 2 From Lemma 5.6, we have the following corollary: Corollary 5.7. We have rank m(D) = c(D). 6. Linearly independent diagrams By Theorem 1.1, each warping matrix M (P ) of a knot projection P has (c(P ) + 1) linearly independent rows. We say that knot diagrams obtained from the same knot projection P are linearly independent if the corresponding rows in M (P ) are linearly independent. We have the following theorem. Theorem 6.1. Let D be a knot diagram with c crossings. Then D and all the c diagrams obtained from D by a single crossing change are linearly independent. Proof. Let m(D) be the warping incidence matrix of D, and vi be the ith row of m(D). As mentioned in Section 5, the warping degree sequence s(D) of D is obtained by v 1 + v 2 + · · · + v c . Let Di be the knot diagram obtained from D by a crossing change at vi . By Corollary 5.3, we have s(Di ) = v 1 + v 2 + · · · + v i−1 + (1 − v i ) + v i+1 + · · · + v c . We will prove that s(D), s(D1 ), s(D2 ), . . . and s(Dc ) are linearly independent. By subtracting s(D) from the others, it is sufficient to show that (v1 +v 2 +· · ·+v c ), (1 −2v 1 ), (1 − 2v 2 ), . . . and (1 − 2v c ) are linearly independent. Let λo (v 1 + v 2 + · · · + v c ) + λ1 (1 − 2v 1 ) + · · · + λc (1 − 2v c ) = 0. Then we have (λ0 − 2λ1 )v 1 + (λ0 − 2λ2 )v 2 + · · · + (λ0 − 2λc )v c + (λ1 + λ2 + · · · + λc )1 = 0. By Lemma 5.6, we have λ0 − 2λ1 = λ0 − 2λ2 = · · · = λ0 − 2λc = λ1 + λ2 + · · · + λc = 0. Hence λ0 = λ1 = · · · = λc = 0. 2
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Acknowledgements The second author expresses gratitude to Professors Kazuaki Kobayashi and Takako Kodate for valuable discussions and suggestions. The authors are also grateful to Yukari Fuseda, Yui Onoda and Professor Yoshiro Yaguchi for valuable discussions. They also thank the referee for careful reading and the helpful suggestions. The second author is partially supported by a Grant for Basic Science Research Projects from the Sumitomo Foundation (No. 150027). References [1] Z. Cheng, A polynomial invariant of virtual knots, Proc. Am. Math. Soc. 142 (2014) 713–725. [2] Z. Cheng, H. Gao, A polynomial invariant of virtual links, J. Knot Theory Ramif. 22 (2013) 1341002, 33 pp. [3] M. Chmutov, T. Hulse, A. Lum, P. Rowell, Plane and spherical curves: an investigation of their invariants, in: Proceedings of the Research Experiences for Undergraduates Program in Mathematics, Oregon State University, 2006, pp. 1–92. [4] T. Fukunaga, The warping degree of a nanoword, Discrete Math. 313 (2013) 599–604. [5] R. Higa, Y. Nakanishi, S. Sato, T. Yamamoto, Crossing information and warping polynomials about the trefoil knot, J. Knot Theory Ramif. 21 (2012) 1250117, 11 pp. [6] Y.H. Im, S. Kim, D.S. Lee, The parity writhe polynomial for virtual knots and flat virtual knots, J. Knot Theory Ramif. 22 (2013) 1250133, 20 pp. [7] A. Kawauchi, Knot theory for spatial graphs attached to a surface, Contemp. Math. (2016), in press. [8] A. Kawauchi, Lectures on Knot Theory, Kyoritsu shuppan Co. Ltd, 2007 (in Japanese). [9] A. Kawauchi, On transforming a spatial graph into a plane graph, Prog. Theor. Phys. Suppl. 191 (2011) 225–234. [10] A. Kawauchi, A. Shimizu, Quantization of the crossing number of a knot diagram, Kyungpook Math. J. 55 (2015) 741–752. [11] A. Kawauchi, K. Yoshida, Topology of prion proteins, J. Math. Syst. Sci. 2 (2012) 237–248. [12] J. Kreinbihl, R. Martin, C. Murphy, M. Renn, J. Townsend, Region almost alternating knots, preprint. [13] A. Lowrance, Alternating distances of knots and links, Topol. Appl. 182 (2015) 53–70. [14] A. Shimizu, The warping degree of a knot diagram, J. Knot Theory Ramif. 19 (2010) 849–857. [15] A. Shimizu, The warping degree of a link diagram, Osaka J. Math. 48 (2011) 209–231. [16] A. Shimizu, The warping matrix of a knot diagram, arXiv:1508.03425, Contemp. Math. (2016), in press. [17] A. Shimizu, The warping polynomial of a knot diagram, J. Knot Theory Ramif. 21 (2012) 1250124, 15 pp.
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
The rank of a warping matrix Taira Akiyama a , Ayaka Shimizu b,∗ , Ryohei Watanabe a a
Department of Information and Computer Engineering, Gunma National College of Technology, 580 Toriba-cho, Maebashi-shi, Gunma, 371-8530, Japan b Department of Mathematics, Gunma National College of Technology, 580 Toriba-cho, Maebashi-shi, Gunma, 371-8530, Japan
a r t i c l e
i n f o
Article history: Received 25 August 2015 Received in revised form 15 March 2016 Accepted 5 April 2016 Available online 12 April 2016 MSC: 57M25
a b s t r a c t The warping matrix has been defined for knot projections and knot diagrams by using warping degrees, and the warping matrix of a knot diagram represents the knot diagram uniquely. In this paper we show that the rank of a warping matrix is one greater than the crossing number of the knot projection or diagram. We also discuss the linearly independence of knot diagrams by considering the warping incidence matrices. © 2016 Elsevier B.V. All rights reserved.
Keywords: Knot diagram Warping degree Warping matrix
1. Introduction In [8], Kawauchi introduced the notion of warping degrees of oriented knot and link diagrams. In [14] and [15], the properties of the warping degrees of knot and link diagrams were studied. For example, inequalities of the warping degree and the crossing number of a knot and link diagram characterizing alternating diagrams were given there. Warping degrees of spatial graphs, virtual knots and nanowords are also studied (see, for example, [4,7,9,11]). In [17] and [10], the warping polynomial and the warping crossing polynomial of a knot diagram were defined via the warping degree labeling and the warping degree sequence which have the information of warping degrees of the diagram, and studied (see also [5]) and applied to define polynomial invariants for virtual knots (see [1,2,6]). The span of a warping polynomial and a warping degree sequence was also studied in [17] and applied to define an alternating distance in [13]. (Recently, it was shown in [12] that every knot has a diagram such that the span is two.) In [16], the warping matrix of a * Corresponding author. E-mail address: [email protected] (A. Shimizu). http://dx.doi.org/10.1016/j.topol.2016.04.003 0166-8641/© 2016 Elsevier B.V. All rights reserved.
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229
Fig. 1. d(Da ) = d(Da ) + 1 and d(Db ) = d(Db ) − 1.
knot projection and a knot diagram was defined by the second author, which has the information of warping degrees and warping polynomials, and it is shown that there is a one-to-one correspondence between oriented knot diagrams on S 2 and warping matrices. Hence the warping matrix can be used as a notation for oriented knots. However, it is not easy to calculate warping matrices for knot projections and diagrams with a large crossing number; the size of the warping matrix of a knot projection with c crossings is 2c × 2c, and that of a knot diagram is (2c − 1) × 2c. One of the motivations on the study of warping matrices is to reduce the size of the matrix. The following theorem would be helpful if we can approach this matter with the theory of matrix decompositions such as the LDU decomposition: Theorem 1.1. Let P be an oriented knot projection on S 2 , and M (P ) the warping matrix of P . We have the following equality: rank M (P ) = c(P ) + 1, where c(P ) is the crossing number of P . Let D be an oriented knot diagram on S 2 , and c(D) the crossing number of D. Let M (D) be the warping matrix of D without signs, which is mentioned concretely in Section 2. We also have the following theorem: Theorem 1.2. We have rank M (D) = c(D) + 1. The rest of this paper is organized as follows: In Section 2, we review the warping matrix. In Section 3, we prove Theorems 1.1 and 1.2. In Section 4, we discuss warping matrices without signs for the trivial knots and prime alternating knots. In Section 5, we define a new matrix, the warping incidence matrix of a knot diagram. In Section 6, we investigate the linearly independence for knot diagrams. 2. Warping matrix In this section we review the warping degree and warping matrix. See [14] and [16] for details. In this paper, we assume that all knot projections and diagrams are oriented. Let D be a knot diagram on S 2 . Take a base point of D avoiding crossing points. We denote by Db the pair of D and b. A crossing point p is said to be a warping crossing point of Db if we meet p as an undercrossing first (and an overcrossing later) when we travel D from b (we encounter each crossing twice). The warping degree d(Db ), which was defined by Kawauchi, of Db is defined to be the number of the warping crossing points of Db . As shown in [14], we have the following: Proposition 2.1. (Lemma 2.5 in [14]) When a base point is moved through an overcrossing (resp. undercrossing) on a knot diagram with the orientation, the warping degree increases (resp. decreases) by one. That is, we have d(Da ) = d(Da ) + 1 and d(Db ) = d(Db ) − 1 for a knot diagram D with the base points illustrated in Fig. 1.
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Fig. 2. Warping matrix.
A knot projection or diagram is said to be trivial if it has no crossings. Let P be a nontrivial knot projection on S 2 with c crossing points. Take a base point at each edge of P , where an edge means a part of P between two crossings which has no crossings in the interior. Label them b1, b2 , . . . , b2c from an edge following the orientation of the diagram. From P , we obtain 2c knot diagrams by giving over/under c information at each crossing of P . We call them D1 , D2 , . . . , D2 . The warping matrix M (P ) of P is the 2c × 2c matrix defined by: M (P ) = ( aij ), where aij = d(Dbi j ). An example is shown in Fig. 2. We consider warping matrices up to permutations on rows and cyclic permutations on columns. Each row of M (P ) is said to be the warping degree sequence of the corresponding diagram. As mentioned in Proposition 2.2 in [16], warping matrices of knot projections have the following properties: • On each row, the difference of two elements which are next to each other is one. c • On each column, n appears times (n = 0, 1, 2, . . . , c). n The ou matrix U (P ) of P is the matrix obtained from M (P ) by the multiplication U (P ) = M (P ) × A, where A is the 2c × 2c matrix as follows: ⎛
−1 ⎜ 1 ⎜ ⎜ ⎜ 0 A=⎜ ⎜ .. ⎜ . ⎜ ⎝ 0 0 We give an example.
0 −1 1 .. . 0 0
... ... ... .. . ... ...
0 0 0 .. . −1 1
⎞ 1 0 ⎟ ⎟ ⎟ 0 ⎟ .. ⎟ ⎟. . ⎟ ⎟ 0 ⎠ −1
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231
Fig. 3. Knot projection P .
Example 2.2. The warping matrix M (P ) of the knot projection P depicted in Fig. 3 is ⎛
0 ⎜1 ⎜ M (P ) = ⎜ ⎝1 2
1 2 0 1
2 1 1 0
⎞ 1 2⎟ ⎟ ⎟ 0⎠ 1
and the ou matrix of P is: ⎛
01 ⎜1 2 ⎜ U (P ) = ⎜ ⎝1 0 21 ⎛
⎞⎛ −1 1 ⎜ ⎟ 2⎟⎜ 1 ⎟⎜ 0⎠⎝ 0 0 1
2 1 1 0
1 ⎜ 1 ⎜ =⎜ ⎝ −1 −1
1 −1 1 −1
−1 1 −1 1
0 0 −1 0 1 −1 0 1 ⎞ −1 −1 ⎟ ⎟ ⎟. 1 ⎠ 1
⎞ 1 0 ⎟ ⎟ ⎟ 0 ⎠ −1
Each row of U (P ) corresponds to a knot diagram, each column corresponds to a crossing, and each element represents the over/under information where 1 means over and −1 means under. Since we pass each crossing twice, there are just c pairs of columns uniquely such that the sum of them is 0. For example, U (P ) of Example 2.2 has the pairs 1st and 4th, and 2nd and 3rd. From the pairing, we can recover the Gauss diagram of P . For details, see [16]. We define the warping matrix without signs for knot diagrams. Let D be an oriented knot diagram on S 2 , and P the knot projection obtained from D by ignoring the over/under information. We define the warping matrix M (D) to be the matrix acquired from M (P ) by deleting the row of D, which is the warping degree sequence of D. We can also obtain M (D) from the ordinary warping matrix M (D) just by removing the signs of elements. For example, we have ⎛
M(
0 ⎜1 ⎜1 ⎜ )=⎜ ⎜2 ⎜2 ⎝ 2 3
1 0 2 1 1 3 2
2 1 3 0 2 2 1
3 2 2 1 1 1 0
2 3 1 2 2 0 1
⎞ 1 2⎟ 0⎟ ⎟ 3⎟ ⎟ 1⎟ ⎠ 1 2
(cf. Fig. 2). 3. Proof of Theorems 1.1 and 1.2 In this section we prove Theorems 1.1 and 1.2. We first show the following lemma: Lemma 3.1. Let c be an integer which is greater than 1. Let a1 , a2 , . . . , ac , ac+1 be integers. We have the following equation.
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232
a1
a
2
.
..
ac−1
ac
ac+1
−1 1 .. . 1 1 −1
1 −1 .. . 1 1 −1
... ... .. . ... ... ...
1 1 .. . −1 1 −1
c
= −2c−1 ai + (c − 2)ac+1 .
1
i=1
−1
−1
1 1
(1)
Proof. By adding the (c + 1)th row to the other rows, the equation (1) is equivalent to:
a1 + ac+1
a +a c+1
2
..
.
ac−1 + ac+1
ac + ac+1
ac+1
−2 0 .. . 0 0 −1
0 −2 .. . 0 0 −1
... ... .. . ... ... ...
0 0 .. . −2 0 −1
c
= −2c−1 ai + (c − 2)ac+1 .
0
i=1
−2
−1
0 0
We will show the equation (2) by an induction on c. • For c = 2, we have
a + a −2 0
3
1
a2 + a3 0 −2 = −2(a1 + a2 ).
a3 −1 −1
Hence (2) holds for c = 2. • Assume (2) holds for c = k − 1 for an integer k greater than two. Now we consider the case c = k:
a1 + ak+1
a +a k+1
2
..
.
ak−1 + ak+1
ak + ak+1
ak+1
−2 0 .. . 0 0 −1
0 −2 .. . 0 0 −1
... ... .. . ... ... ...
0 0 .. . −2 0 −1
.
0
−2
−1
0 0
The Laplace expansion along the (k + 1)th column yields:
a1 + ak+1 −2 0 . . . 0
a +a 0 −2 . . . 0
k+1
2
.. .. .. . .
(−1)k+(k+1) × (−2)
. . . .
ak−1 + ak+1 0 0 . . . −2
−1 −1 . . . −1
ak+1
a1 + ak+1 −2 0 . . . 0
a +a 0 −2 . . . 0
k+1
2
.. .. .. . .
. + (−1)(k+1)+(k+1) × (−1)
. . . .
ak−1 + ak+1 0 0 . . . −2
ak + ak+1 0 0 . . . 0
Laplace expansion again along the kth row at the second term yields:
(2)
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a1 + ak+1
a +a k+1
2
.
.. 2
ak−1 + ak+1
ak+1
−2 0 .. . 0 −1
0 −2 .. . 0 −1
... ... .. . ... ...
−2
0
− (−1)k+1 (ak + ak+1 ) .
.
.
−2
0 −1
0 0
0 −2 .. . 0
... ... .. . ...
0
0
..
. .
−2
233
(3)
We remark that the (k, 1) element of the first matrix in (3) is ak+1 . By assumption, (3) is equal to 2 −2
(k−1)−1
= −2
k−1
k−1
k−1
ai + (k − 1 − 2)ak+1
i=1
ai + (k − 3)ak+1
− 2k−1 (ak + ak+1 )
i=1
= −2
k−1
k−1
= −2
k−1
ai + (k − 3)ak+1 + ak + ak+1
i=1 k
− (−1)k+1 (ak + ak+1 )(−2)k−1
ai + (k − 2)ak+1
.
i=1
Hence (2) holds. 2 By multiplying by −1 the (c + 1)th column, we obtain the following corollary from Lemma 3.1: Corollary 3.2. We have
a1
a
2
.
..
ac−1
ac
ac+1
−1 1 .. . 1 1 −1
1 −1 .. . 1 1 −1
... ... .. . ... ... ...
1 1 .. . −1 1 −1
−1
−1
c
c−1
=2 ai + (c − 2)ac+1 .
−1
i=1
1
1
(4)
Now we prove Theorem 1.1. Proof of Theorem 1.1. For a 2c × 2c matrix M (P ) = (a1 , a2 , . . . , a2c ), subtract a2c−1 from a2c , a2c−2 from a2c−1 , a2c−3 from a2c−2 , . . . , and a1 from a2 . Let (a, v 1 , v 2 , . . . , v 2c−1 ) be the matrix obtained by the procedure. Note that (v 1 , v 2 , . . . , v 2c−1 ) is a submatrix of U (P ). By the property of ou matrix, there are just (c − 1) pairs of columns v i and v j uniquely such that v i + v j = 0. For each pair, add one to the other. By reordering some columns v i , we have (a, v ϕ1 , v ϕ2 , . . . , v ϕc , 0, 0, . . . , 0). Hence rank M (P ) ≤ c(P ) + 1. Next we show that a, v ϕ1 , v ϕ2 , . . . , v ϕc are linearly independent. Since the columns v ϕ1 , v ϕ2 , . . . , v ϕc correspond to all the c crossings, and the rows correspond to all the over/under information, we can obtain
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the following (c + 1) × (c + 1) submatrix of (a, v ϕ1 , v ϕ2 , . . . , v ϕc ) by reordering some rows ⎛
−1 1 .. . 1 1 −1
a1 ⎜ a ⎜ 2 ⎜ . ⎜ .. ⎜ ⎜ ⎜ ac−1 ⎜ ⎝ ac ac+1
1 −1 .. . 1 1 −1
... ... .. . ... ... ...
1 1 .. . −1 1 −1
⎞ 1 1 ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ 1 ⎟ ⎟ −1 ⎠ −1
where ai is an element of a (i = 1, 2, . . . , c + 1). By Lemma 3.1, the determinant of the submatrix is −2
c
c−1
ai + (c − 2)ac+1
.
i=1
Remark that the elements of a are all non-negative and there are at most one 0. Hence the determinant is non-zero. Therefore a, v ϕ1 , v ϕ2 , . . . , v ϕc are linearly independent. Hence rank M (P ) = c(P ) + 1.
2
We prove Theorem 1.2. Proof of Theorem 1.2. Similar to Theorem 1.1 except that M (D) has one row fewer than M (P ). Remark that we can obtain at least one of the following two submatrices ⎛
a1 ⎜ a ⎜ 2 ⎜ . ⎜ .. ⎜ ⎜ ⎜ ac−1 ⎜ ⎝ ac ac+1
−1 1 .. . 1 1 −1
1 −1 .. . 1 1 −1
... ... .. . ... ... ...
1 1 .. . −1 1 −1
⎛ ⎞ a1 1 ⎜ ⎟ 1 ⎟ ⎜ a2 ⎜ . ⎟ ⎜ . ⎟ ⎟ and ⎜ . ⎜ ⎟ ⎜ ac−1 1 ⎟ ⎜ ⎟ ⎝ ac −1 ⎠ −1 ac+1
which are the matrices of Lemma 3.1 and Corollary 3.2.
−1 1 .. . 1 1 −1
1 −1 .. . 1 1 −1
... ... .. . ... ... ...
1 1 .. . −1 1 −1
⎞ −1 −1 ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ −1 ⎟ ⎟ 1 ⎠ 1
2
4. Warping matrices for the trivial knot and prime alternating knots In this section we investigate warping degrees and warping matrices and discuss the warping matrices for the trivial knot and prime alternating knots. We first show the following lemma: Lemma 4.1. Let Nc = 0
c c c c +1 +2 + ··· + c 0 1 2 c
for a positive integer c. Then we have Nc = 2c−1 c. Proof. From the binomial theorem, we have c c c c 2 = + + + ··· + . 0 1 2 c c
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When c is odd, we have 2c = 2
c c c c , + + + · · · + c−1 0 1 2 2
and then we have c c c c + + + · · · + c−1 = 2c−1 . 0 1 2 2 Similarly, when c is even, we have c c c c c + c , 2 =2 + + + ··· + c − 1 0 1 2 2 2 c
and then we have c c c c 1 c + + + ... c = 2c−1 − . 0 1 2 2 2c 2 −1 Hence we have c c c c Nc = c + + + · · · + c−1 0 1 2 c = c × 2c−1 for an odd integer c, and c c c c c c + Nc = c + + + ··· + c 2 2c 0 1 2 2 −1 c c 1 c = c 2c−1 − + = 2c−1 c 2 2c 2 2c for an even integer c. 2 We have the following proposition: Proposition 4.2. Let M (P ) be the warping matrix of a knot projection P with c crossings. We have (1 1 . . . 1)M (P ) = (2c−1 c 2c−1 c . . . 2c−1 c), where (1 1 . . . 1) is the row vector with length 2c whose elements are all 1, and (2c−1 c 2c−1 c . . . 2c−1 c) is the row vector with length 2c whose elements are all 2c−1 c. Proof. The row vector (1 1 . . . 1)M (P ) represents the sum of all the rows of M (P ). As we have seen in Section 2 (and [10,16]), n appears nc times at each column of M (P ) (n = 0, 1, 2, . . . , c). Hence each element of (1 1 . . . 1)M (P ) is c c c c +1 +2 + ··· + c 0 1 2 c
0
and this is equal to 2c−1 c by Lemma 4.1.
2
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Fig. 4. The knot diagram D of the trivial knot has the warping degree sequence 2123432.
Fig. 5. A connected sum P Q of P and Q.
Fig. 6. A Gauss diagram represents a connected sum if and only if the chords can be divided into two sides, where the connecting edges separate the chords into two sides.
Since (1 1 . . . 1)M (D) also represents the sum of all the rows of M (D), we can reobtain the warping degree sequence of D with c crossings from M (D) by (2c−1 c 2c−1 c . . . 2c−1 c) − (1 1 . . . 1)M (D). By the definition of the warping degree, we can say that a knot diagram D represents the trivial knot if D has zero in the warping degree sequence. Note that the converse does not hold (see, for example, Fig. 4). In terms of matrices, we have the following proposition: Proposition 4.3. Let M (D) be a 2c−1 × 2c warping matrix without signs. If (2c−1 c 2c−1 c . . . 2c−1 c) − (1 1 . . . 1)M (D) has zero as an element, then M (D) represents the trivial knot. Next we discuss about a connected sum on knot projections and diagrams. For two knot projections (resp. diagrams) P and Q, we say that the knot projection (resp. diagram) depicted in the right hand side in Fig. 5 is a connected sum of P and Q, and denote it by P Q. We say a knot projection (resp. diagram) is prime if it is not any connected sum of two nontrivial knot projections (resp. diagrams). In this paper, we call the two edges of P Q connecting P and Q the connecting edges of P and Q. It is known that a Gauss diagram represents a connected sum if and only if the chords of the Gauss diagram can be divided into two sides as shown in Fig. 6 (see, for example, [3]). Connecting edges have the same warping degrees as follows: Proposition 4.4. Let D = P Q be a knot diagram which is a connected sum of P and Q. Let e1 , e2 be the connecting edges of P and Q and let b1 , b2 be a base point on e1 , e2 , respectively. Then we have d(Db1 ) = d(Db2 ). Proof. From e1 to e2 , we go through every crossing of just P (or Q) twice; as an overcrossing once and as an undercrossing once. Hence Db1 and Db2 have the same warping degree because of the property of the warping degree in Proposition 2.1. 2
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Fig. 7. A chord has e and f at the opposite sides.
Moreover, we have the following proposition: Proposition 4.5. A warping matrix M (P ) is a warping matrix of a knot projection which is a connected sum of two nontrivial knot projections if and only if there exists a pair of the same columns in M (P ). Proof. If a knot projection P is a connected sum, the connecting edges have the same warping degree with any over/under information at the crossings of P by Proposition 4.4. Hence the corresponding columns in M (P ) have all the same elements. Next we will show that if M (P ) has a pair of the same columns then the corresponding edges are connecting edges by showing the contraposition. Let P be a knot projection and let e, f be a pair of edges of P which is not any pair of connecting edges. Since the chords of the Gauss diagram of P cannot be separated by e and f , there exists a chord which has e and f at the opposite sides in the Gauss diagram (see Fig. 7). We call v the corresponding crossing of P . Now we will show that we can give over/under information to the crossings of P so that e and f have different warping degrees when we take base points there. Let D be a knot diagram obtained from P by giving over/under information at each crossing. If e and f have the same warping degrees, apply a crossing change at the crossing v. Then, as shown in Theorem 1 (ii) in [10], the warping degrees with base points on the edges from the overcrossing of v to the undercrossing of v decreases by one, and the warping degrees with base points on the edges from the undercrossing to overcrossing increases by one. Then the warping degrees of e and f differ by the crossing change. Hence the columns corresponding to e and f have different elements in M (P ). 2 For M (D), we have the following: Corollary 4.6. A warping matrix without signs M (D) is a warping matrix without signs of a knot diagram which is a connected sum of two nontrivial knot diagrams if and only if there exists a pair of the same columns in M (D). As mentioned in [17], the warping degree sequence of an alternating knot is also alternating like (k k + 1 k . . . k + 1). It is also shown in [17] that a knot diagram D is alternating if and only if the span of the warping degree sequence is one, where the span of a warping degree sequence is the difference of the maximal entry and the minimal entry in the sequence. We have the following corollary: Corollary 4.7. A 2c−1 × 2c warping matrix without signs M (D) represents a minimal crossing diagram of a prime alternating knot with the crossing number c (= rank M (D) − 1) if and only if • M (D) does not have any pair of the same columns, and • the span of (2c−1 c 2c−1 c . . . 2c−1 c) − (1 1 . . . 1)M (D) is one. 5. Warping incidence matrix In this section we define the warping incidence matrix for knot diagrams. Let D be a knot diagram on S 2 with c crossings. Label the edges e1 , e2 , . . . , e2c from an edge following the orientation of the diagram. Label the crossings v1 , v2 , . . . , vc . The warping incidence matrix m(D) of D is defined as follows:
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Fig. 8. Warping incidence matrix.
Fig. 9. A crossing and four edges.
m(D) = (aij ),
where aij =
1 (vi is a warping crossing point of Dbj ); 0 (vi is not a warping crossing point of Dbj ),
where bj is a base point on ej . An example is shown in Fig. 8. We consider warping incidence matrices up to permutations on rows and cyclic permutations on columns. By definition, we have the following: Proposition 5.1. The sum of all the rows of the warping incidence matrix of a knot diagram D is the warping degree sequence of D. We show the following proposition. Proposition 5.2. Let m(D) = (aij ) be the warping incidence matrix of a knot diagram D with c(D) = c. For any integer k ∈ {1, 2, . . . , c}, there exists a unique integer l ∈ {1, 2, . . . , 2c} such that ⎧ ⎪ ⎨ ak l−1 = 0, ak l = 1 and ⎪ ⎩a i l−1 = ai l (i = k).
(5)
Proof. See Fig. 9. At the kth row in m(D), 1 appears from lth column through mth column because the corresponding edges have the crossing vk as a warping crossing point, whereas 0 appears from (m + 1)th column through (l − 1)th column. Hence we have ak l−1 = 0 and ak l = 1. Since the two edges el−1 and el have the same warping crossing points except at vk , we have ai l−1 = ai l for i = k. 2 We have the following corollaries: Corollary 5.3. Let D be a knot diagram and let Di be the diagram obtained from D by a crossing change at the crossing vi . Then m(Di ) is obtained from m(D) by switching 0 and 1 at the row corresponding to vi . Corollary 5.4. We can obtain the Gauss diagram of D without signs from m(D). We also define the signed warping incidence matrix of a knot diagram to be the matrix obtained from the warping incidence matrix by replacing “1” with “−1” or “1” at each row corresponding to a negative crossing. Then we have the following corollary:
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Corollary 5.5. We can obtain the Gauss diagram and the knot diagram on S 2 from a signed warping incidence matrix. Next we show the following lemma: Lemma 5.6. Let D be a knot diagram with c(D) = c. Let m(D) be the warping incidence matrix of D, and v i the ith row of m(D) (i = 1, 2, . . . , c). Then v 1 , v 2 , . . . , v c and 1 are linearly independent, where 1 is the row vector (11 . . . 1) with length 2c. Proof. Let λ1 v 1 + λ2 v 2 + · · · + λc v c + λc+1 1 = 0.
(6)
For each k ∈ {1, 2, . . . , c}, there exists an integer l satisfying (5) in Proposition 5.2. Since the lth and (l−1)th elements of (6) are zero, the difference of them, which is λk , is also zero. Hence we have λ1 = λ2 = · · · = λc = 0 and therefore λc+1 = 0. 2 From Lemma 5.6, we have the following corollary: Corollary 5.7. We have rank m(D) = c(D). 6. Linearly independent diagrams By Theorem 1.1, each warping matrix M (P ) of a knot projection P has (c(P ) + 1) linearly independent rows. We say that knot diagrams obtained from the same knot projection P are linearly independent if the corresponding rows in M (P ) are linearly independent. We have the following theorem. Theorem 6.1. Let D be a knot diagram with c crossings. Then D and all the c diagrams obtained from D by a single crossing change are linearly independent. Proof. Let m(D) be the warping incidence matrix of D, and vi be the ith row of m(D). As mentioned in Section 5, the warping degree sequence s(D) of D is obtained by v 1 + v 2 + · · · + v c . Let Di be the knot diagram obtained from D by a crossing change at vi . By Corollary 5.3, we have s(Di ) = v 1 + v 2 + · · · + v i−1 + (1 − v i ) + v i+1 + · · · + v c . We will prove that s(D), s(D1 ), s(D2 ), . . . and s(Dc ) are linearly independent. By subtracting s(D) from the others, it is sufficient to show that (v1 +v 2 +· · ·+v c ), (1 −2v 1 ), (1 − 2v 2 ), . . . and (1 − 2v c ) are linearly independent. Let λo (v 1 + v 2 + · · · + v c ) + λ1 (1 − 2v 1 ) + · · · + λc (1 − 2v c ) = 0. Then we have (λ0 − 2λ1 )v 1 + (λ0 − 2λ2 )v 2 + · · · + (λ0 − 2λc )v c + (λ1 + λ2 + · · · + λc )1 = 0. By Lemma 5.6, we have λ0 − 2λ1 = λ0 − 2λ2 = · · · = λ0 − 2λc = λ1 + λ2 + · · · + λc = 0. Hence λ0 = λ1 = · · · = λc = 0. 2
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