• 検索結果がありません。

On the arc index of knots and links

N/A
N/A
Protected

Academic year: 2022

シェア "On the arc index of knots and links"

Copied!
82
0
0

読み込み中.... (全文を見る)

全文

(1)

On the arc index of knots and links

Hwa Jeong Lee

KAIST

May 22, 2014

Intelligence ofLow-DimensionalTopology RIMS, Kyoto University, Japan

(2)

Table of Contents

1 Short Survey

Arc presentation and arc index Representations of an arc presentation History and Known results

2 New Results (joint with Hideo Takioka) Arc index of Kanenobu knots

Arc index of cable links

(3)

Table of Contents

1 Short Survey

Arc presentation and arc index Representations of an arc presentation History and Known results

2 New Results (joint with Hideo Takioka) Arc index of Kanenobu knots

Arc index of cable links

(4)

Arc presentation and arc index

Anarc presentationof a knot or a linkLis an embedding ofLcontained in the union of finitely many half planes, calledpages, with a common boundary line, calledbinding axis, in such a way that each half plane contains a properly embedded single simple arc.

H1

H5 5 4 3 2 1

H1 5 4 3 2 1

H2 5 4 3 2 1

H3 5 4 3 2 1

H4 5 4 3 2 1

H5 5 4 3 2 1

The minimum number of pages among all arc presentations of a linkLis called thearc indexofLand is denoted byα(L).

(5)

Links with arc index up to 5

α(L) 2 3 4 5

L unknot none 2-component unlink, Hopf link trefoil

(6)

Representations of an arc presentation

1,3 2,5

1,4 3,5

2,4

1 2

3 4

5 C

5 4 3 2 1

®4

®2

®3

®1

®5

®4

®2

®3

®1

®5

®4

®2

®3 ®1

®5

®4

®2

®3

®1

®5

(7)

History

? Brunn(1897) proved that any link has a diagram with only one multiple point (not necessarily double).

? Birman-Menasco(1994) used arc presentations of companion knots to study the braid index of their satellites.

? Cromwell(1995) used the term “arc index” and established some of its basic properties.

? Dynnikov(2006) proved that any arc-presentation of the unknot admits a monotonic simplification by elementary moves : this yields a simple algorithm for recognizing the unknot.

(8)

Grid diagram

Cromwell, 1995

1. Every link admits an arc presentation.

2. If a nonsplit linkLis a connected sum of two linksL1andL2, then α(L1]L2)=α(L1)+α(L2)−2.

• Every link admits a grid diagram.

• A grid diagram gives rise to an arc presentation and vice versa.

a b

c

d e

grid diagram

(9)

Grid diagram

Cromwell, 1995

1. Every link admits an arc presentation.

2. If a nonsplit linkLis a connected sum of two linksL1andL2, then α(L1]L2)=α(L1)+α(L2)−2.

• Every link admits a grid diagram.

• A grid diagram gives rise to an arc presentation and vice versa.

a b

c

d e

grid diagram a

b c

d e

(10)

Grid diagram

Cromwell, 1995

1. Every link admits an arc presentation.

2. If a nonsplit linkLis a connected sum of two linksL1andL2, then α(L1]L2)=α(L1)+α(L2)−2.

• Every link admits a grid diagram.

• A grid diagram gives rise to an arc presentation and vice versa.

(11)

Elementary Moves on Grid Diagrams

Dynnikov, 2006

Two grid diagrams of the same link can be obtained from each other by a finite sequence of the following elementary moves.

• stabilizationanddestabilization;

• interchanging neighbouring edges if their pairs of endpoints do not interleave;

• cyclic permutation ofvertical(horizontal) edges.

(12)

Elementary Moves on Grid Diagrams

Dynnikov, 2006

Two grid diagrams of the same link can be obtained from each other by a finite sequence of the following elementary moves.

• stabilization and destabilization;

• interchanging neighbouring edges if their pairs of endpoints do not interleave;

• cyclic permutation ofvertical(horizontal) edges.

(13)

Elementary Moves on Grid Diagrams

Dynnikov, 2006

Two grid diagrams of the same link can be obtained from each other by a finite sequence of the following elementary moves.

• stabilization and destabilization;

• interchanging neighbouring edges if their pairs of endpoints do not interleave;

• cyclic permutation ofvertical(horizontal) edges.

(14)

Elementary Moves on Grid Diagrams

Dynnikov, 2006

Two grid diagrams of the same link can be obtained from each other by a finite sequence of the following elementary moves.

• stabilization and destabilization;

• interchanging neighbouring edges if their pairs of endpoints do not interleave;

• cyclic permutation ofvertical(horizontal) edges.

(15)

The front projection of a Legendrian knot

A knot diagramDrepresentsthe front projection of a Legendrian knotif (1) Dhas no vertical tangencies,

(2) the only non-smooth points are generalized cusps, and

(3) at each crossing the slope of the strand with the overcrossing is smaller than with the undercrossing.

(16)

Thurston-Bennequin number

? A grid diagram gives rise to a Legendrian knot and vice versa.

+45˚

w#@#06 c#@#6

For a grid diagramG,Thurston-Bennequin numberis defined by tb(G)=w(G)−c(G)

wherew(G) andc(G) are the writhe and the number of southeast corners of G, respectively.

Themaximal Thurston-Bennequin numberof a knotK, written tb(K), is the maximal tb over all grid diagrams forK.

(17)

Thurston-Bennequin number

? A grid diagram gives rise to a Legendrian knot and vice versa.

+45˚

For a grid diagramG,Thurston-Bennequin numberis defined by tb(G)=w(G)−c(G)

wherew(G) andc(G) are the writhe and the number of southeast corners of G, respectively.

Themaximal Thurston-Bennequin numberof a knotK, written tb(K), is the maximal tb over all grid diagrams forK.

(18)

A relation between α(K ) and tb(K)

Matsuda, 2006

−α(K)≤tb(K)+tb(K), whereKis the mirror image of a knotK.

Question(Ng), 2012

Does a grid diagram realizingα(K) of a knotKnecessarily realizetb(K)?

An equivalent statement is that

−α(K)=tb(K)+tb(K) for any knotK.

(19)

Known Results I

• [Beltrami, 2002] Arc index for prime knots up to 10 crossings are determined.

• [Ng, 2006] Arc index for prime knots up to 11 crossings are determined.

? [Nutt, 1999] All knots up to arc index 9 are identified.

? [Jin et al., 2006] All prime knots up to arc index 10 are identified.

? [Jin-Park, 2010] All prime knots up to arc index 11 are identified.

? [Jin-Kim] All prime knots up to arc index 12 are identified.(preprint)

(20)

Known Results I

• [Beltrami, 2002] Arc index for prime knots up to 10 crossings are determined.

• [Ng, 2006] Arc index for prime knots up to 11 crossings are determined.

? [Nutt, 1999] All knots up to arc index 9 are identified.

? [Jin et al., 2006] All prime knots up to arc index 10 are identified.

? [Jin-Park, 2010] All prime knots up to arc index 11 are identified.

? [Jin-Kim] All prime knots up to arc index 12 are identified.(preprint)

(21)

Kau ff man polynomial F

L

(a, z)

TheKauffman polynomialof an oriented knot or linkLis defined by FL(a,z)=a−w(D)ΛD(a,z)

whereDis a diagram ofL,w(D) the writhe ofDandΛD(a,z) the polynomial determined by the rules (K1), (K2) and (K3).

(K1) ΛO(a,z)=1 whereOis the trivial knot diagram.

(K2) ΛD+(a,z)+ ΛD(a,z)=z(ΛD0(a,z)+ ΛD(a,z)).

(K3) aΛD(a,z)= ΛD(a,z)=a−1ΛD (a,z).

D+ D D0 D D D D

(22)

α(L) ≥ spread

a

(F

L

) + 2.

Thea-spreadof the Kauffman polynomialFL(a,z)=a−w(D)ΛD(a,z) is denoted byspreada(FL)and defined by the formula

spreada(FL)=max-dega(FL)−min-dega(FL).

Morton-Beltrami, 1998 LetLbe a link. Then

α(L)≥spreada(FL)+2.

Notice :spreada(FL)=spreadaD)for any diagramDofL.

(23)

Wheel diagram

• Bae-Park described an algorithm which transforms a diagram of a non-split link withncrossings into a wheel diagram with at mostn+2 spokes.

1,3

2,5

1,4 3,5

2,4

5 4 3 2 1

Bae-Park, 2000

IfLis a non-split link, thenα(L)≤c(L)+2.

(24)

The idea of Bae-Park Theorem

Bae-Park, 2000

IfLis a non-split link, thenα(L)≤c(L)+2.

3 3 2

2 4 4

3 3 2

2,4 4

1 1

3 2

2,4 1,3

1,4

2,5 3,5

2,4 1,3

1,4

Idea : The sum of number of regions and spokes is unchanged.

(25)

A relation between α(L) and c(L)

• L:non-split alternating link=⇒α(L)=c(L)+2.

◦ [Morton-Beltrami, 1998] For any linkL,α(L)≥spreada(FL(a,z))+2.

◦ [Thistlethwaite, 1988] IfLis an alternating link, spreada(FL(a,z))≥c(L).

◦ [Bae-Park, 2000] IfLis a non-split link, then α(L)≤c(L)+2.

• L:nonalternating prime=⇒spreada(FL(a,z))+2≤α(L)≤c(L).

◦ [M-B] For any linkL,α(L)≥spreada(FL(a,z))+2.

◦ [Jin-Park, 2010] A prime linkLis nonalternatingif and only if α(L)≤c(L).

(26)

A relation between α(L) and c(L)

• L:non-split alternating link=⇒α(L)=c(L)+2.

◦ [Morton-Beltrami, 1998] For any linkL,α(L)≥spreada(FL(a,z))+2.

◦ [Thistlethwaite, 1988] IfLis an alternating link, spreada(FL(a,z))≥c(L).

◦ [Bae-Park, 2000] IfLis a non-split link, then α(L)≤c(L)+2.

• L:nonalternating prime=⇒spreada(FL(a,z))+2≤α(L)≤c(L).

◦ [M-B] For any linkL,α(L)≥spreada(FL(a,z))+2.

◦ [Jin-Park, 2010] A prime linkLis nonalternatingif and only if α(L)≤c(L).

(27)

Known Results III

? [Etnyre-Honda, 2001]α(Tp,q)=|p|+|q|

? [L-Jin] Arc index of pretzel knots of type (−p,q,r) (submitted)

? [L] Arc index of Montesinos links of type (−r1,r2,r3) (preprint)

? [L-Takioka] On the arc index of Kanenobu knots

? [L-Takioka] On the arc index of cable links

(28)

Known Results III

? [Etnyre-Honda, 2001]α(Tp,q)=|p|+|q|

? [L-Jin] Arc index of pretzel knots of type (−p,q,r) (submitted)

? [L] Arc index of Montesinos links of type (−r1,r2,r3) (preprint)

? [L-Takioka] On the arc index of Kanenobu knots

? [L-Takioka] On the arc index of cable links

(29)

Arc index of Kanenobu knots

(30)

What are Kanenobu Knots?

p

q

K(p, q)

n > 0 n < 0

n

n =0

=

T. Kanenobu,Infinitely many knots with the same polynomial invariant, Proc. Amer. Math. Soc. 97 (1986) 158–162.

T. Kanenobu,Examples on polynomial invariants of knots and links, Math. Ann.275(1986) 555–572.

Kanenobu, 1986

K(p,q)=K(q,p) and K(p,q)=K(−p,−q) .

(31)

K(p, q) with | p | ≤ q

p q

It is sufficient to considerK(p,q) with

|p| ≤qin order to determine the arc index ofK(p,q).

? K(p,q)=K(q,p).

? K(p,q)=K(−p,−q)=K(−q,−p).

? α(L)=α(L).

(32)

K(p, q) with | p | ≤ q

p q

q

It is sufficient to considerK(p,q) with

|p| ≤qin order to determine the arc index ofK(p,q).

? K(p,q)=K(q,p).

? K(p,q)=K(−p,−q)=K(−q,−p).

? α(L)=α(L).

(33)

Main results

Theorem K1

Let 1≤p≤q and pq≥3. Then α(K(p,q))=p+q+6.

Theorem K2

Letp=0 and q≥3. Then q+6≤α(K(0,q))≤q+7.

Theorem K3

Letp=−1 and q≥3. Then q+5≤α(K(−1,q))≤q+7.

Theorem K4

Letp=−2 and q≥3. Then q+4≤α(K(−2,q))≤q+7.

(34)

Theorem K2. Let p = 0, q ≥ 3.

Then q + 6 ≤ α(K(0, q)) ≤ q + 7.

K DT Name b+6 α(K) b+7

K(0,3)K(0,−3) 11n50 9 10 10

K(0,4)K(0,−4) 12n145 10 11 11 K(0,5)K(0,−5) 13n579 11 11 12 K(0,6)K(0,−6) 14n2459 12 12\ 13

Jin et al.,Prime knots with arc index up to 10, Series on Knots and Everything Book vol. 40, World Scientific Publishing Co., 6574, 2006.

Jin-Park,A tabulation of prime knots up to arc index 11, JKTR vol. 20, No. 11, pp. 1537–1635.

\ Jin-Kim,Prime knots with arc index 12 up to 16 crossings, preprint.

(35)

Theorem K3. Let p = − 1, q ≥ 3.

Then q + 5 ≤ α(K ( − 1, q)) ≤ q + 7.

K DT Name b+5 α(K) b+7

K(−1,3)K(1,−3) 11n37 8 10 10

K(−1,4)K(1,−4) 12n414 9 11 11 K(−1,5)K(1,−5) 13n2036 10 11 12 K(−1,6)K(1,−6) 14n9271 11 12 13 K(−1,7)K(1,−7) 15n46855 12 12 14

(36)

Theorem K4. Let p = − 2, q ≥ 3.

Then q + 4 ≤ α(K ( − 2, q)) ≤ q + 7.

K DT Name b+4 α(K) b+7

K(−2,3)K(2,−3) 13n1836 7 10 10 K(−2,4)K(2,−4) 14n11995 8 11 11 K(−2,5)K(2,−5) 15n54616 9 11 12 K(−2,6)K(2,−6) 16n331702 10 12 13

(37)

Arc index of cable links

(38)

What is a satellite knot?

B V =S1×D2⊂S3: a standard solid torus.

B K1: a knot embedded inVs.t. every meridinal disk ofVintersectsK1. B N(K) : a tubular neighborhood of a knotK.

Lethbe a faithful homeomorphism fromVontoN(K).

V N(K) K1

h

The imageh(K1) is called asatellite knotwithcompanion knot K.

(39)

Cable links and Whitehead doubles

Letp,qbe integers withp>0.

V N(K) K1

h

K1 h(K1) denoted by

(p,q)-torus linkTp,q (p,q)-cable link K(p,q)

t positive Whitehead double K(+,t)

t negative Whitehead double K(−,t) ,

t=2 t=-2

(40)

A standard diagram of cable links

B D: a diagram of a knotKwith an (1,1)-tangleT.

B w(D) : the writhe ofD.

B βp:=σ1σ2· · ·σp−1andb:=βq−pw(D)p .

i i +1 i i +1

¾i ¾i-1

ThenSD(p,q)is a diagram ofK(p,q)whereTpis ap-strand parallel tangle ofT as shown in the figure below. We callS(p,q)D thestandard diagramofK(p,q) obtained from Dand the braidb(p,q−pw(D))-twist.

p strands

D S

T T

b

(p,q) p

D

(41)

Canonical arc index

LetGbe a grid diagram of a knotKandp,qintegers withp>0.

The grid diagram obtained by the canonical (p,q)-cabling algorithm ofGis called thecanonical grid diagramofK(p,q)obtained from Gand denoted by G(p,q).

Letα(G(p,q)) denote the number of vertical line segments ofG(p,q). The canonical arc indexofK(p,q), denoted byαc(K(p,q)), is defined as follows:

αc(K(p,q))=min{α(G(p,q))|Gis a grid diagram ofK}.

Note:α(K(p,q))≤αc(K(p,q)).

Question 1

α(K(p,q))=αc(K(p,q))?

(42)

Canonical arc index

LetGbe a grid diagram of a knotKandp,qintegers withp>0.

The grid diagram obtained by the canonical (p,q)-cabling algorithm ofGis called thecanonical grid diagramofK(p,q)obtained from Gand denoted by G(p,q).

Letα(G(p,q)) denote the number of vertical line segments ofG(p,q). The canonical arc indexofK(p,q), denoted byαc(K(p,q)), is defined as follows:

αc(K(p,q))=min{α(G(p,q))|Gis a grid diagram ofK}.

Note:α(K(p,q))≤αc(K(p,q)).

Question 1

α(K(p,q))=αc(K(p,q))?

(43)

K α(K(2,q)) α(K(+,t)) α(K(−,t)) 31 −q+12 if q1 −2t+13 if t0 −2t+14 if t1

10 if 2q12 11 if 1t5 11 if 2t6

q2 if q13 2t if t6 2t1 if t7

41 −q+6 if q≤ −7 −2t+7 if t≤ −4 −2t+8 if t≤ −3

12 if −6q6 13 if −3t2 13 if −2t3

q+6 if q7 2t+8 if t3 2t+7 if t4

51 −q+20 if q5 −2t+21 if t2 −2t+22 if t3

14 if 6q20 15 if 3t9 15 if 4t10

q6 if q21 2t4 if t10 2t5 if t11

52 −q2 if q≤ −17 −2t1 if t≤ −9 −2t if t≤ −8 14 if −16q≤ −2 15 if −8t≤ −2 15 if −7t≤ −1

q+16 if q≥ −1 2t+18 if t≥ −1 2t+17 if t0

61 −q+6 if q≤ −11 −2t+7 if t≤ −6 −2t+8 if t≤ −5

16 if −10q6 17 if −5t2 17 if −4t3

q+10 if q7 2t+12 if t3 2t+11 if t4

(44)

K α(K(2,q)) α(K(+,t)) α(K(−,t)) 62 −q+14 if q≤ −3 −2t+15 if t≤ −2 −2t+16 if t≤ −1

16 if −2q14 17 if −1t6 17 if 0t7

q+2 if q15 2t+4 if t7 2t+3 if t8

63 −q+8 if q≤ −9 −2t+9 if t≤ −5 −2t+10 if t≤ −4

16 if −8q8 17 if −4t3 17 if −3t4

q+8 if q9 2t+10 if t4 2t+9 if t5

71 −q+28 if q9 −2t+29 if t4 −2t+30 if t5

18 if 10q28 19 if 5t13 19 if 6t14

q10 if q29 2t8 if t14 2t9 if t15

72 −q2 if q≤ −21 −2t1 if t≤ −11 −2t if t≤ −10 18 if −20q≤ −2 19 if −10t≤ −2 19 if −9t≤ −1

q+20 if q≥ −1 2t+22 if t≥ −1 2t+21 if t0

73 −q+24 if q5 −2t+25 if t2 −2t+26 if t3

18 if 6q24 19 if 3t11 19 if 4t12

q6 if q25 2t4 if t12 2t5 if t13

(45)

K α(K(2,q)) α(K(+,t)) α(K(−,t)) 74 −q2 if q≤ −21 −2t1 if t≤ −11 −2t if t≤ −10

18 if −20q≤ −2 19 if −10t≤ −2 19 if −9t≤ −1

q+20 if q≥ −1 2t+22 if t≥ −1 2t+21 if t0

75 −q+24 if q5 −2t+25 if t2 −2t+26 if t3

18 if 6q24 19 if 3t11 19 if 4t12

q6 if q25 2t4 if t12 2t5 if t13

76 −q+2 if q≤ −17 −2t+3 if t≤ −9 −2t+4 if t≤ −8

18 if −16q2 19 if −8t0 19 if −7t1

q+16 if q3 2t+18 if t1 2t+17 if t2

77 −q+8 if q≤ −11 −2t+9 if t≤ −6 −2t+10 if t≤ −5

18 if −10q8 19 if −5t3 19 if −4t4

q+10 if q9 2t+12 if t4 2t+11 if t5

81 −q+6 if q≤ −15 −2t+7 if t≤ −8 −2t+8 if t≤ −7

20 if −14q6 21 if −7t2 21 if −6t3

q+14 if q7 2t+16 if t3 2t+15 if t4

(46)

K α(K(2,q)) α(K(+,t)) α(K(−,t)) 82 −q+22 if q1 −2t+23 if t0 −2l+24 if t1

20 if 2q22 21 if 1t10 21 if 2t11

q2 if q23 2t if t11 2l1 if t12

83 −q+10 if q≤ −11 −2t+11 if t≤ −6 −2l+12 if t≤ −5 20 if −10q10 21 if −5t4 21 if −4t5

q+10 if q11 2t+12 if t5 2l+11 if t6

84 −q+14 if q≤ −7 −2t+15 if t≤ −4 −2l+16 if t≤ −3 20 if −6q14 21 if −3t6 21 if −2t7

q+6 if q15 2t+8 if t7 2l+7 if t8

85 −q+22 if q1 −2t+23 if t0 −2l+24 if t1

20 if 2q22 21 if 1t10 21 if 2t11

q2 if q23 2t if t11 2l1 if t12

86 −q+18 if q≤ −3 −2t+19 if t≤ −2 −2l+20 if t≤ −1

20 if −2q18 21 if −1t8 21 if 0t9

q+2 if q19 2t+4 if t9 2l+3 if t10

(47)

K α(K(2,q)) α(K(+,t)) α(K(−,t)) 87 −q+16 if q≤ −5 −2t+17 if t≤ −3 −2t+18 if t≤ −2

20 if −4q16 21 if −2t7 21 if −1t8

q+4 if q17 2t+6 if t8 2t+5 if t9

88 −q+8 if q≤ −13 −2t+9 if t≤ −7 −2t+10 if t≤ −6 20 if −12q8 21 if −6t3 21 if −5t4

q+12 if q9 2t+14 if t4 2t+13 if t5

89 −q+10 if q≤ −11 −2t+11 if t≤ −6 −2t+12 if t≤ −5 20 if −10q10 21 if −5t4 21 if −4t5

q+10 if q11 2t+12 if t5 2t+11 if t6

810 −q+16 if q≤ −5 −2t+17 if t≤ −3 −2t+18 if t≤ −2 20 if −4q16 21 if −2t7 21 if −1t8

q+4 if q17 2t+6 if t8 2t+5 if t9

811 −q+2 if q≤ −19 −2t+3 if t≤ −10 −2t+4 if t≤ −9 20 if −18q2 21 if −9t0 21 if −8t1

q+18 if q3 2t+20 if t1 2t+19 if t2

(48)

K α(K(2,q)) α(K(+,t)) α(K(−,t)) 812 −q+10 if q≤ −11 −2t+11 if t≤ −6 −2t+12 if t≤ −5

20 if −10q10 21 if −5t4 21 if −4t5

q+10 if q11 2t+12 if t5 2t+11 if t6

813 −q+8 if q≤ −13 −2t+9 if t≤ −7 −2t+10 if t≤ −6 20 if −12q8 21 if −6t3 21 if −5t4

q+12 if q9 2t+14 if t4 2t+13 if t5

814 −q+18 if q≤ −3 −2t+19 if t≤ −2 −2t+20 if t≤ −1

20 if −2q18 21 if −1t8 21 if 0t9

q+2 if q19 2t+4 if t9 2t+3 if t10

815 −q+26 if q5 −2t+27 if t2 −2t+28 if t3

20 if 6q26 21 if 3t12 21 if 4t13

q6 if q27 2t4 if t13 2t5 if t14

816 −q+16 if q≤ −5 −2t+17 if t≤ −3 −2t+18 if t≤ −2 20 if −4q16 21 if −2t7 21 if −1t8

q+4 if q17 2t+6 if t8 2t+5 if t9

(49)

K α(K(2,q)) α(K(+,t)) α(K(−,t)) 817 −q+10 if q≤ −11 −2t+11 if t≤ −6 −2t+12 if t≤ −5

20 if −10q10 21 if −5t4 21 if −4t5

q+10 if q11 2t+12 if t5 2t+11 if t6

818 −q+10 if q≤ −11 −2t+11 if t≤ −6 −2t+12 if t≤ −5 20 if −10q10 21 if −5t4 21 if −4t5

q+10 if q11 2t+12 if t5 2t+11 if t6

819 −q+24 if q9 −2t+25 if t4 −2t+26 if t5 14 if 10q24 15 if 5t11 15 if 6t12

q10 if q25 2t8 if t12 2t9 if t13

820 −q+4 if q≤ −13 −2t+5 if t≤ −7 −2t+6 if t≤ −6 16 if −12q4 17 if −6t1 17 if −5t2

q+12 if q5 2t+14 if t2 2t+13 if t3

821 −q+18 if q1 −2t+19 if t0 −2t+20 if t1

16 if 2q18 17 if 1t8 17 if 2t9

q2 if q19 2t if t9 2t1 if t10

(50)

Questions

Question 1

α(K(p,q))=αc(K(p,q))?

Question 2

For two minimal grid diagramsG,G0of a knotK, we have α(G(p,q))=α(G0(p,q))?

Question 3

IfGis a minimal grid diagram of a knotK, then we have αc(K(p,q))=α(G(p,q))?

(51)

Terminology

A corner ofGis calledseif it is a SouthEast corner, and similarlysw, neor nwwhen SouthWest, NorthEast or NorthWest, respectively.

? ne(G) : the number ofnecorners ofG

? se(G) : the number ofsecorners ofG

? tb(G)=w(G)−se(G).

90˚

crossing change

G G*

Note: ne(G)=se(G)and se(G)=ne(G).

(52)

Terminology

A corner ofGis calledseif it is a SouthEast corner, and similarlysw, neor nwwhen SouthWest, NorthEast or NorthWest, respectively.

? ne(G) : the number ofnecorners ofG

? se(G) : the number ofsecorners ofG

? tb(G)=w(G)−se(G).

90˚

crossing change

G G*

Note: ne(G)=se(G) and se(G)=ne(G).

(53)

Canonical (p, q)-cabling algorithm

We start a grid diagramGof a knotK.

B Letvbe a point on the rightmost vertical line segment which is not any corner.

Step I : Identify each corner ofG

B LetCGdenote the sequence of oriented indexed corners ofGwhere the indexiindicates theith meeting corner when we travel alongGstarting fromvaway from the rightmostne-corner.

ne8 ne4

ne10 nw7

sw9

sw6

sw2 se5

se1 nw3

v

¯ ¯

¯

¯

¯

¯

¯

¯

¯

¯

CG: (se1↓,sw2↑,nw3↑,ne4↓,se5↓,sw6↑,nw7↑,ne8↓,sw9↓,ne10↓) B LetC+GandCGbe the subsequences ofCGwhich is made up of all

sw,ne-corners and allse,nw-corners ofCG, respectively.

C+G: (sw2↑,ne4↓,sw6↑,ne8↓,sw9↓,ne10↓), CG: (se1↓,nw3↑,se5↓,nw7↑).

(54)

Canonical (p, q)-cabling algorithm

Step II : Obtain the standard diagramS(p,q)G

Given two integersp>0 andq, we haveS(p,q)G ofK(p,q)fromG.

(3,5)-twist (3,0)-twist

ne8 ne4

ne10 nw7

sw9 sw6

sw2 se5

se1 nw3

v

¯ ¯

¯

¯

¯

¯ ¯

¯

¯

¯

SG(3,14) SG(3,9)

G =G(3,9)

¯35 ¯30

w(G)=3 βq−pw(G)p53 βq−pw(G)p03 Ifq=pw(G) thenS(p,pw(G))G is the canonical grid diagram ofK(p,pw(G)). Suppose thatq−pw(G),0.

(55)

Canonical (p, q)-cabling algorithm

(3,5)-twist ne8

ne4 ne10 nw7

sw9 sw6

sw2

se5 se1

nw3

v

¯ ¯

¯

¯

¯

¯ ¯

¯

¯

¯

SG(3,14) G

sw(3)2¯

¯35

We denote

B the corners corresponding tose,sw,nwandnebyse(p),sw(p),nw(p)andne(p), respectively.

B the sequences corresponding toCG,C+GandCGbyCS(p,q) G

,C+

S(p,q)G andC

S(p,q)G in S(p,q)G , respectively,

CS(3,14) G

: (se(3)1,sw(3)2,nw(3)3,ne(3)4,se(3)5,sw(3)6,nw(3)7,ne(3)8,sw(3)9,ne(3)10), C+

S(3,14)G : (sw(3)2,ne(3)4,sw(3)6,ne(3)8,sw(3)9,ne(3)10), C

S(3,14)G : (se(3)1↓,nw(3)3↑,se(3)5↓,nw(3)7↑).

(56)

Canonical (p, q)-cabling algorithm

Step III (1) : Define the canonical grid form of(p,q−pw(G))-twist Agrid formof ap-strand braid is a diagram such that each strand of the braid is expressed as a union of vertical and horizontal line segments and at each crossing the vertical line segment crosses over the horizontal line segment.

Letβp1σ2· · ·σp−1. Applying the rule of the figure below to theβ±1p

p-1

1 2 3 p-1p 1 2 3 p-1p 1 2 3 p-1p 1 2 3 p-1p

we can obtain a grid form ofβspfor an integerswhich is called thecanonical grid formofβsp.

参照

関連したドキュメント

Keywords: Convex order ; Fréchet distribution ; Median ; Mittag-Leffler distribution ; Mittag- Leffler function ; Stable distribution ; Stochastic order.. AMS MSC 2010: Primary 60E05

Let X be a smooth projective variety defined over an algebraically closed field k of positive characteristic.. By our assumption the image of f contains

2 Combining the lemma 5.4 with the main theorem of [SW1], we immediately obtain the following corollary.. Corollary 5.5 Let l &gt; 3 be

In Section 3, we show that the clique- width is unbounded in any superfactorial class of graphs, and in Section 4, we prove that the clique-width is bounded in any hereditary

Inside this class, we identify a new subclass of Liouvillian integrable systems, under suitable conditions such Liouvillian integrable systems can have at most one limit cycle, and

The main problem upon which most of the geometric topology is based is that of classifying and comparing the various supplementary structures that can be imposed on a

In this article we prove the following result: if two 2-dimensional 2-homogeneous rational vector fields commute, then either both vector fields can be explicitly integrated to

Two surface-links in R 4 are equivalent if and only if their marked graph diagrams can be transformed into each other by a finite sequence of 8 types of moves, called the