Tabulation of the clasp number of prime knots with up to 10 crossings - PowerPoint PPT Presentation

. . 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) 結び目の数学 VI （日大文理） December 20, 2013 Kengo Kawamura (Osaka City Univ.) December 20, 2013 1 / 26 結び目の数学 VI （日大文理）

Today’s Talk . .. Introduction 1 The clasp number of a knot Properties Main Result Table of the clasp number of prime knots up to 10 crossings . .. Characterizing of the Alexander module via the clasp disk 2 The Alexander inv. obtained from a knot K with c ( K ) ≤ n Application Proof of Main result Kengo Kawamura (Osaka City Univ.) December 20, 2013 2 / 26 結び目の数学 VI （日大文理）

Today’s Talk . .. Introduction 1 The clasp number of a knot Properties Main Result Table of the clasp number of prime knots up to 10 crossings . .. Characterizing of the Alexander module via the clasp disk 2 The Alexander inv. obtained from a knot K with c ( K ) ≤ n Application Proof of Main result Kengo Kawamura (Osaka City Univ.) December 20, 2013 3 / 26 結び目の数学 VI （日大文理）

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 D c ( D ) : the clasp number of K . Kengo Kawamura (Osaka City Univ.) December 20, 2013 4 / 26 結び目の数学 VI （日大文理）

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]) ⇒ c ( K p,q ) = ( | p | − 1)( | q | − 1) . . K p,q : the ( p, q ) -torus knot = . . 2 ([K. Morimoto ’89]) . . c ( K ) = 1 ⇐ . ⇒ K is a doubled knot. (cf. [T. Kobayashi ’89]) Kengo Kawamura (Osaka City Univ.) December 20, 2013 5 / 26 結び目の数学 VI （日大文理）

Proof of max { g ( K ) , u ( K ) } ≤ c ( K ) Sketch Proof. g ( K ) ≤ c ( K ) u ( K ) ≤ c ( K ) � Kengo Kawamura (Osaka City Univ.) December 20, 2013 6 / 26 結び目の数学 VI （日大文理）

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.) December 20, 2013 7 / 26 結び目の数学 VI （日大文理）

knot knot knot g u c g u c g u c 3 1 1 1 1 8 1 1 1 1 8 15 2 2 2 4 1 1 1 1 8 2 3 2 3 8 16 3 2 3 5 1 2 2 2 8 3 1 2 2 8 17 3 1 3 5 2 1 1 1 8 4 2 2 2 8 18 3 2 3 6 1 1 1 1 8 5 3 2 3 8 19 3 3 3 6 2 2 1 2 8 6 2 2 2 8 20 2 1 2 6 3 2 1 2 8 7 3 1 3 8 21 2 1 2 7 1 3 3 3 8 8 2 2 2 9 1 4 4 4 7 2 1 1 1 8 9 3 1 3 9 2 1 1 1 7 3 2 2 2 8 10 3 2 3 9 3 3 3 3 7 4 1 2 2 8 11 2 1 2 9 4 2 2 2 7 5 2 2 2 8 12 2 2 2 9 5 1 2 2 7 6 2 1 2 8 13 2 1 2 9 6 3 3 3 7 7 2 1 2 8 14 2 1 2 9 7 2 2 2 Kengo Kawamura (Osaka City Univ.) December 20, 2013 8 / 26 結び目の数学 VI （日大文理）

knot g u c knot g u c knot g u c 9 8 2 2 2 9 22 3 1 3 9 36 3 2 3 9 9 3 3 3 9 23 2 2 2 9 37 2 2 2 9 10 2 3 3 9 24 3 1 3 9 38 2 3 3 9 11 3 2 3 9 25 2 2 2 9 39 2 1 X 9 12 2 1 2 9 26 3 1 3 9 40 3 2 3 9 13 2 3 3 9 27 3 1 3 9 41 2 2 X 9 14 2 1 2 9 28 3 1 3 9 42 2 1 2 9 15 2 2 2 9 29 3 2 3 9 43 3 2 3 9 16 3 3 3 9 30 3 1 3 9 44 2 1 2 9 17 3 2 3 9 31 3 2 3 9 45 2 1 2 9 18 2 2 2 9 32 3 2 3 9 46 1 2 2 9 19 2 1 2 9 33 3 1 3 9 47 3 2 3 9 20 3 2 3 9 34 3 1 3 9 48 2 2 2 9 21 2 1 2 9 35 1 3 3 9 49 2 3 3 ( X = 2 or 3) Kengo Kawamura (Osaka City Univ.) December 20, 2013 9 / 26 結び目の数学 VI （日大文理）

knot g u c knot g u c knot g u c 10 1 1 1 1 10 15 3 2 3 10 29 3 2 3 10 2 4 3 4 10 16 2 2 10 30 2 1 X X 10 3 1 2 2 10 17 4 1 4 10 31 2 1 2 10 4 2 2 2 10 18 2 1 2 10 32 3 1 3 10 5 4 2 4 10 19 3 2 3 10 33 2 1 X 10 6 3 3 3 10 20 2 2 2 10 34 2 2 2 10 7 2 1 2 10 21 3 2 3 10 35 2 2 2 10 8 3 2 3 10 22 3 2 3 10 36 2 2 2 10 9 4 1 4 10 23 3 1 3 10 37 2 2 2 10 10 2 1 2 10 24 2 2 2 10 38 2 2 2 10 11 2 X X 10 25 3 2 3 10 39 3 2 3 10 12 3 2 3 10 26 3 1 3 10 40 3 2 3 10 13 2 2 2 10 27 3 1 3 10 41 3 2 3 10 14 3 2 3 10 28 2 2 X 10 42 3 1 3 ( X = 2 or 3) Kengo Kawamura (Osaka City Univ.) December 20, 2013 10 / 26 結び目の数学 VI （日大文理）

knot g u c knot g u c knot g u c 10 43 3 2 3 10 57 3 2 3 10 71 3 1 3 10 44 3 1 3 10 58 2 2 2 10 72 3 2 3 10 45 3 2 3 10 59 3 1 3 10 73 3 1 3 10 46 4 3 4 10 60 3 1 3 10 74 2 2 X 10 47 4 X 4 10 61 3 X 3 10 75 3 2 3 10 48 4 2 4 10 62 4 2 4 10 76 3 X 3 10 49 3 3 3 10 63 2 2 2 10 77 3 X 3 10 50 3 2 3 10 64 4 2 4 10 78 3 2 3 10 51 3 3 10 65 3 2 3 10 79 4 4 X X 10 52 3 2 3 10 66 3 3 3 10 80 3 3 3 10 53 2 3 3 10 67 2 2 2 10 81 3 2 3 10 54 3 X 3 10 68 2 2 X 10 82 4 1 4 10 55 2 2 2 10 69 3 2 3 10 83 3 2 3 10 56 3 2 3 10 70 3 2 3 10 84 3 1 3 ( X = 2 or 3) Kengo Kawamura (Osaka City Univ.) December 20, 2013 11 / 26 結び目の数学 VI （日大文理）

knot g u c knot g u c knot g u c 10 85 4 2 4 10 99 4 2 4 10 113 3 1 3 10 86 3 2 3 10 100 4 4 10 114 3 1 3 X 10 87 3 2 3 10 101 2 3 3 10 115 3 2 3 10 88 3 1 3 10 102 3 1 3 10 116 4 2 4 10 89 3 2 3 10 103 3 3 3 10 117 3 2 3 10 90 3 2 3 10 104 4 1 4 10 118 4 1 4 10 91 4 1 4 10 105 3 2 3 10 119 3 1 3 10 92 3 2 3 10 106 4 2 4 10 120 2 3 3 10 93 3 2 3 10 107 3 1 3 10 121 3 2 3 10 94 4 2 4 10 108 3 2 3 10 122 3 2 3 10 95 3 1 3 10 109 4 2 4 10 123 4 2 4 10 96 3 2 3 10 110 3 2 3 10 124 4 4 4 10 97 2 2 3 10 111 3 2 3 10 125 3 2 3 10 98 3 2 3 10 112 4 2 4 10 126 3 2 3 ( X = 2 or 3) Kengo Kawamura (Osaka City Univ.) December 20, 2013 12 / 26 結び目の数学 VI （日大文理）

knot g u c knot g u c knot g u c 10 127 3 2 3 10 141 3 1 3 10 155 3 2 3 10 128 3 3 3 10 142 3 3 3 10 156 3 1 3 10 129 2 1 10 143 3 1 3 X 10 157 3 2 3 10 130 2 2 10 144 2 2 2 X 10 158 3 2 3 10 131 2 1 X 10 145 2 2 2 10 159 3 1 3 10 132 2 1 2 10 146 2 1 2 10 160 3 2 3 10 133 2 1 2 10 147 2 1 2 10 161 3 3 3 10 134 3 3 3 10 148 3 2 3 10 162 2 2 X 10 135 2 2 2 10 149 3 2 3 10 163 3 2 3 10 136 2 1 2 10 150 3 2 3 10 164 2 1 X 10 137 2 1 2 10 151 3 2 3 10 165 2 2 X 10 138 3 2 3 10 152 4 4 4 10 139 4 4 4 10 153 3 2 3 10 140 2 2 2 10 154 3 3 3 ( X = 2 or 3) Kengo Kawamura (Osaka City Univ.) December 20, 2013 13 / 26 結び目の数学 VI （日大文理）

Today’s Talk . .. Introduction 1 The clasp number of a knot Properties Main Result Table of the clasp number of prime knots up to 10 crossings . .. Characterizing of the Alexander module via the clasp disk 2 The Alexander inv. obtained from a knot K with c ( K ) ≤ n Application Proof of Main result Kengo Kawamura (Osaka City Univ.) December 20, 2013 14 / 26 結び目の数学 VI （日大文理）

Characterizing of the Alexander module via the clasp disk Kengo Kawamura (Osaka City Univ.) December 20, 2013 15 / 26 結び目の数学 VI （日大文理）

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.) December 20, 2013 16 / 26 結び目の数学 VI （日大文理）

The Alexander inv. obtained from a knot K w/ c ( K ) ≤ n Example: Kengo Kawamura (Osaka City Univ.) December 20, 2013 17 / 26 結び目の数学 VI （日大文理）

The Alexander inv. obtained from a knot K w/ c ( K ) ≤ n Example: Kengo Kawamura (Osaka City Univ.) December 20, 2013 18 / 26 結び目の数学 VI （日大文理）

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 1 n ε 1 O . . ... ... . . W := and U :=  .     . .    a n 1 · · · a nn O ε n ✓ ✏ Theorem 1. ([K. Morimoto ’98]) K : a knot with c ( K ) ≤ n . A = tV − V T : the Alexander matrix. ( tW − W T ) Then, A is equivalent to ( t − 1) U − tI . ✒ ✑ Kengo Kawamura (Osaka City Univ.) December 20, 2013 19 / 26 結び目の数学 VI （日大文理）

Proof of Theorem 1 ( W ) Proof. O V = . − I − U ( tW − W T ) I tV − V T A = = − 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.) December 20, 2013 20 / 26 結び目の数学 VI （日大文理）