Tabulation of the clasp number of prime knots with up to 10 crossings
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1 . Tabulation of the clasp number of prime knots with up to 10 crossings... Kengo Kawamura (Osaka City University) joint work with Teruhisa Kadokami (East China Normal University).. VI December 20, 2013 Kengo Kawamura (Osaka City Univ.) VI December 20, / 26
2 Today s Talk...1 Introduction The clasp number of a knot Properties Main Result Table of the clasp number of prime knots up to 10 crossings...2 Characterizing of the Alexander module via the clasp disk The Alexander inv. obtained from a knot K with c(k) n Application Proof of Main result Kengo Kawamura (Osaka City Univ.) VI December 20, / 26
3 Today s Talk...1 Introduction The clasp number of a knot Properties Main Result Table of the clasp number of prime knots up to 10 crossings...2 Characterizing of the Alexander module via the clasp disk The Alexander inv. obtained from a knot K with c(k) n Application Proof of Main result Kengo Kawamura (Osaka City Univ.) VI December 20, / 26
4 The clasp number of a knot Fact: Any (oriented) knot K S 3 bounds a clasp disk D. c(d) := the number of clasp singularities of D. c(k) := min c(d): the clasp number of K. D Kengo Kawamura (Osaka City Univ.) VI December 20, / 26
5 Properties. max{g(k), u(k)} c(k). ([T. Shibuya 74]). c(k 1 #K 2 ) 3 = c(k 1 #K 2 ) = c(k 1 ) + c(k 2 ). ([K. Morimoto 87] & [H. Matsuda 03]). K p,q : the (p, q)-torus knot = c(k p,q ) = ([K. Morimoto 89]) ( p 1)( q 1) 2.. c(k) = 1 K is a doubled knot. (cf. [T. Kobayashi 89]) Kengo Kawamura (Osaka City Univ.) VI December 20, / 26
![Matsuda 03]). K p,q : the (p, q)-torus knot = c(k p,q ) = ([K.](/docs-images/53/20129733/images/page_5.jpg "Morimoto 89]) ( p 1)( q 1) 2.. c(k) = 1 K is a doubled knot. (cf. [T.")
6 Proof of max{g(k), u(k)} c(k) Sketch Proof. g(k) c(k) u(k) c(k) Kengo Kawamura (Osaka City Univ.) VI December 20, / 26
7 Main Result Question:?K: a knot s.t. max{g(k), u(k)} < c(k). Answer/Main Result: max{g(10 97 ), u(10 97 )} < c(10 97 ). Note: g(10 97 ) = 2 and u(10 97 ) = 2. ([Y. Miyazawa 98], [Y. Nakanishi 05]) Kengo Kawamura (Osaka City Univ.) VI December 20, / 26
![Note: g(10 97 ) = 2 and u(10 97 ) = 2. ([Y. Miyazawa 98], [Y.](/docs-images/53/20129733/images/page_7.jpg "Nakanishi 05]) Kengo Kawamura (Osaka City Univ.")
8 knot g u c knot g u c knot g u c Kengo Kawamura (Osaka City Univ.) VI December 20, / 26
9 knot g u c knot g u c (X = 2 or 3) knot g u c X X Kengo Kawamura (Osaka City Univ.) VI December 20, / 26
10 knot g u c X X knot g u c X X (X = 2 or 3) knot g u c X X Kengo Kawamura (Osaka City Univ.) VI December 20, / 26
11 knot g u c X X X knot g u c X X (X = 2 or 3) knot g u c X X X X Kengo Kawamura (Osaka City Univ.) VI December 20, / 26
12 knot g u c knot g u c X (X = 2 or 3) knot g u c Kengo Kawamura (Osaka City Univ.) VI December 20, / 26
13 knot g u c X X X knot g u c (X = 2 or 3) knot g u c X X X Kengo Kawamura (Osaka City Univ.) VI December 20, / 26
14 Today s Talk...1 Introduction The clasp number of a knot Properties Main Result Table of the clasp number of prime knots up to 10 crossings...2 Characterizing of the Alexander module via the clasp disk The Alexander inv. obtained from a knot K with c(k) n Application Proof of Main result Kengo Kawamura (Osaka City Univ.) VI December 20, / 26
15 Characterizing of the Alexander module via the clasp disk Kengo Kawamura (Osaka City Univ.) VI December 20, / 26
16 The Alexander inv. obtained from a knot K w/ c(k) n The sign of a clasp & The homological basis Kengo Kawamura (Osaka City Univ.) VI December 20, / 26
17 The Alexander inv. obtained from a knot K w/ c(k) n Example: Kengo Kawamura (Osaka City Univ.) VI December 20, / 26
18 The Alexander inv. obtained from a knot K w/ c(k) n Example: Kengo Kawamura (Osaka City Univ.) VI December 20, / 26
19 The Alexander inv. obtained from a knot K w/ c(k) n V : the Seifert matrix obtained from the homological basis {[α 1 ], [α 2 ],..., [α n ], [β 1 ], [β 2 ],..., [β n ]}. We put a ij := lk(α i, α + j ). Moreover, a 11 a 1n W :=..... a n1 a nn and U := ε 1 O... O ε n. Theorem 1. ([K. Morimoto 98]) K: a knot with c(k) n. A = tv V T : the Alexander matrix. Then, A is equivalent to (t 1)U ( tw W T) ti. Kengo Kawamura (Osaka City Univ.) VI December 20, / 26
![.., [α n ], [β 1 ], [β 2 ],..., [β n ]}. We put a ij := lk(α i, α + j ). Moreover, a 11 a 1n W :=.](/docs-images/53/20129733/images/page_19.jpg ".... a n1 a nn and U := ε 1 O... O ε n. Theorem 1. ([K. Morimoto 98]) K: a knot with c(k) n.")
20 Proof of Theorem 1 Proof. A = tv V T = ( W O V = I U ). ( tw W T I ) ti (t 1)U ( tw W T I (t 1)U(tW W T ) ti O ( O I (t 1)U(tW W T ) ti O ( (t 1)U(tW W T ) ti O O I ) ) ) (t 1)U(tW W T ) ti. Kengo Kawamura (Osaka City Univ.) VI December 20, / 26
21 The Alexander poly. of a knot K with c(k) 2 m ij : the (i, j)-entry of the matrix (t 1)U ( tw W T) ti. Then, it follows that m ij = { εi a ii (t 1) 2 t (i = j) ε i (t 1)(a ij t a ji ) (i j). Theorem 2. ([K. Morimoto 98]) (1) K: a knot with c(k) 1 = K (t). = b 1 (t 1) 2 + t (b 1 Z). (2) K: a knot with c(k) 2 = K (t). = ( b 1 b 2 + εb 3 (b 3 + δ) ) (t 1) 4 + (b 1 + b 2 εδ)t(t 1) 2 + t 2 (b 1, b 2, b 3 Z, ε = ±1, δ {0, 1}). Kengo Kawamura (Osaka City Univ.) VI December 20, / 26
22 Proof of Theorem 2 (2) Proof. K (t) =. ( ε1 a det 11 (t 1) 2 t ε 1 (t 1)(a 12 t a 21 ) ε 2 (t 1)(a 21 t a 12 ) ε 2 a 22 (t 1) 2 t = ( ε 1 a 11 (t 1) 2 t )( ε 2 a 22 (t 1) 2 t ) ε 1 ε 2 (t 1) 2 (a 12 t a 21 )(a 21 t a 12 ) ) = ε 1 ε 2 ( a11 a 22 a 12 a 21 ) (t 1) 4 + ( ε 1 a 11 ε 2 a 22 + ε 1 ε 2 (a 21 a 12 ) 2) t(t 1) 2 + t 2 b 1 := ε 1 a 11, b 2 := ε 2 a 22, b 3 := a 12, ε := ε 1 ε 2 { 0 (α1 α and δ := a 21 a 12 = 2 = ) 1 (α 1 α 2 ). = ( b 1 b 2 + εb 3 (b 3 + δ) ) (t 1) 4 + (b 1 + b 2 εδ)t(t 1) 2 + t 2. Kengo Kawamura (Osaka City Univ.) VI December 20, / 26
23 Application For any knot K, K (t) = g i=0 c 2i t g i (t 1) 2i (c 0 = 1) K(z) = g c 2i z 2i i=0 (c 0 = 1). Theorem 2. ([K. Morimoto 98]) (2) K: a knot with c(k) 2 = K (z) = ( b 1 b 2 + εb 3 (b 3 + δ) ) z 4 + (b 1 + b 2 εδ)z Corollary 3. (b 1, b 2, b 3 Z, ε = ±1, δ {0, 1}). K (z) = c 4 z 4 + c 2 z = c(k) 3. s.t. c 4 3 (mod 8) and c 2 2 (mod 4) Kengo Kawamura (Osaka City Univ.) VI December 20, / 26
24 Proof of Main Result Main result: max{g(10 97 ), u(10 97 )} < c(10 97 ). (Note: g(10 97 ) = 2 and u(10 97 ) = 2.) Proof. We can see that c(10 97 ) 3. Since 1097 (z) = 5z 4 + 2z 2 + 1, by Corollary 3 c(10 97 ) 3. Therefore, c(10 97 ) = 3. Kengo Kawamura (Osaka City Univ.) VI December 20, / 26
25 Proof of Corollary 3 (1/2) Theorem 2. ([K. Morimoto 98]) (2) K: a knot with c(k) 2 = K (z) = ( b 1 b 2 + εb 3 (b 3 + δ) ) z 4 + (b 1 + b 2 εδ)z Corollary 3. (b 1, b 2, b 3 Z, ε = ±1, δ {0, 1}). K (z) = c 4 z 4 + c 2 z = c(k) 3. s.t. c 4 3 (mod 8) and c 2 2 (mod 4) Proof. Suppose that the Conway polynomial K (z) of a knot K with c(k) 2 satisfies the conditions of c 4 and c 2. By Theorem 2 (2), c 4 = b 1 b 2 + εb 3 (b 3 + δ) and c 2 = b 1 + b 2 εδ. Kengo Kawamura (Osaka City Univ.) VI December 20, / 26
26 Proof of Corollary 3 (2/2) The case of δ = 0 c 4 = b 1 b 2 + εb 2 3 and c 2 = b 1 + b 2. ( ) 2 b1 b 2 d := = c2 2 c 4 + εb εb 2 3 (mod 8). Then d 2, 5, 6 or 7 (mod 8), and d cannot be a square integer. This is a contradiction. The case of δ = 1 c 4 = b 1 b 2 + εb 3 (b 3 + 1) and c 2 = b 1 + b 2 ε. d := (b 1 b 2 ) 2 = (c 2 + ε) 2 4 ( c 4 εb 3 (b 3 + 1) ) 1 4 (mod 8) 5 (mod 8). Then d cannot be a square integer. This is a contradiction. Kengo Kawamura (Osaka City Univ.) VI December 20, / 26
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