On Kauffman polynomial of alternating knot and HOMFLY polynomial of - - PowerPoint PPT Presentation

On Kauffman polynomial of alternating knot and HOMFLY polynomial of its Whitehead double

三浦 嵩広

(神戸大学大学院 理学研究科)

結び目の数理II

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Kauffman polynomial The Kauffman polynomial F(L) = F(L; a, x) ∈ Z[a±1, x±1] is a link invariant defined as follows: F(L) := a−w(D)[D], where w(D) is the writhe of a diagram D of a link L. In this talk, we study the highest degree of x in F(L).

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HOMFLY polynomial The HOMFLY polynomial P(L) = P(L; v, z) ∈ Z[v±1, z±1] is a link invariant defined as follows: W(K) : a Whitehead double of a knot K

➔ ➔

In this talk, we study the highest degree of z in P(W(K)).

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✛ ✚ ✘ ✙

Conj.1 [Kidwell-Stoimenow 2003] K : a nontrivial knot. ⇒ 2(x-maxdegF(K)) + 2 = z-maxdegP(W(K)). In this talk, we discuss the following conjecture.

✛ ✚ ✘ ✙

Conj.2 K : an alternating prime knot. ⇒ 2(x-maxdegF(K)) + 2 = z-maxdegP(W(K)). Prop.1 [Thistlethwaite 1988] K : an alternating prime knot ⇒ x-maxdegF(K) = c(K) − 1, where c(K) is the crossing number of K.

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From this Fact, Conj.2 ⇔ Conj.2’.

✛ ✚ ✘ ✙

Conj.2 K : an alternating prime knot. ⇒ 2(x-maxdegF(K)) + 2 = z-maxdegP(W(K)).

✛ ✚ ✘ ✙

Conj.2’ K : an alternating prime knot. ⇒ z-maxdegP(W(K)) = 2c(K). Remark K : a nontrivial knot ⇒ z-maxdegP(W(K)) ≤ 2c(K),

by Morton’s inequality z-maxdegP(D) ≤ 1 − s(D) + c(D), where s(D) is the number of Seifert circles of a diagram D

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K the coefficient of xc−1 of F(K) the coefficient of z2c of P(W(K)) 31 a−3(a−1 + a) vα(v−1 − v) 41 (a−1 + a) vα(v−1 − v) 51 a−5(a−1 + a) vα(v−1 − v) 52 a−5(a−1 + a) vα(v−1 − v) 61 a−2(a−1 + a) vα(v−1 − v) 62 a−2(a−1 + a) vα(v−1 − v) 63 (a−1 + a) vα(v−1 − v) 71 a−7(a−1 + a) vα(v−1 − v) 72 a−7(a−1 + a) vα(v−1 − v) 73 a−7(a−1 + a) vα(v−1 − v) 74 a−7(a−1 + a) vα(v−1 − v) 75 a−7(a−1 + a) vα(v−1 − v) 76 a−3(a−1 + a) vα(v−1 − v) 77 a(a−1 + a) vα(v−1 − v) ——————————————————α : a certain integer.

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K the coefficient of xc−1 of F(K) the coefficient of z2c of P(W(K)) 81 a−3(a−1 + a) vα(v−1 − v) . . . . . . . . . 816 2a−3(a−1 + a) 2vα(v−1 − v) 817 2a−3(a−1 + a) 2vα(v−1 − v) 818 3a−3(a−1 + a) 3vα(v−1 − v) 91 a−3(a−1 + a) vα(v−1 − v) . . . . . . . . . 939 2a−3(a−1 + a) 2vα(v−1 − v) 940 4a−3(a−1 + a) 4vα(v−1 − v) 941 2a−3(a−1 + a) 2vα(v−1 − v)

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K the coefficient of xc−1 of F(K) the coefficient of z2c of P(W(K)) 81 1a−3(a−1 + a) 1vα(v−1 − v) . . . . . . . . . 816 2a−3(a−1 + a) 2vα(v−1 − v) 817 2a−3(a−1 + a) 2vα(v−1 − v) 818 3a−3(a−1 + a) 3vα(v−1 − v) 91 1a−3(a−1 + a) 1vα(v−1 − v) . . . . . . . . . 939 2a−3(a−1 + a) 2vα(v−1 − v) 940 4a−3(a−1 + a) 4vα(v−1 − v) 941 2a−3(a−1 + a) 2vα(v−1 − v)

Def.

F(K; a) : the coefficient of xc−1 of F(K). P(W(K); v) : the coefficient of z2c of P(W(K)).

ϕ(K):= 1

2F(K; 1).

π(K):= 1

2|P(W(K); i)|.

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Previous research

✛ ✚ ✘ ✙

• Thm. [Gruber 2009]

K : an alternating prime knot. ⇒ ϕ(K) ≡ π(K) (mod 2). Main results Main result 1 K : an alternating prime knot with c(K) ≤ 12 ⇒ ϕ(K) = π(K). Cor. K : an alternating prime knot with c(K) ≤ 12 ⇒ 2(x-maxdegF(K))+2 = z-maxdegP(W(K)).

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Main results Def. The standard projection of the (3, n)-torus link (n = 2, 3, . . .)

this tangle of this projection

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Main result 2 D: a link projection . D′: a knot projection obtained by repeating the following

• perations from D.

➔ ➔ ➔ ➔

K : the knot presented by an alternating diagram with D′ ⇒ ϕ(K) = π(K) = 2|β|2|γ| ∏

n≥2

(n − 1)|δn|, where |β|, |γ|, |δn| is the number of times of (β), (γ), (δn) to obtain D′.

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Example

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Example |β| = 1

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Example |β| = 1, |γ| = 2

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Example |β| = 1, |γ| = 2, |δ4| = 1

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Example |β| = 1, |γ| = 2, |δ4| = 1, ϕ(K) = π(K) = 212231 = 24

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Outline of the proof of Main results D : a connected, link projection. Dalt : an alternating diagram with D.

• Def. of ϕ(D)

ϕ(D) := 1

2F(Dalt; 1),

where F(Dalt; a) the coefficient of xc(D)−1 in F(Dalt; a, x).

• Def. of π(D)

π(D) := 1

2|P(W(Dalt); i)|, where

P(W(Dalt); v) the coefficient of z2c(D) in P(W(Dalt); v, z). e.g. D W(Dalt)

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Outline of the proof of Main results Prop.2 [Thistlethwaite 1988] (ii) Dalt : a reducible diagram ⇒ ϕ(D) = 0 (iii) For a non-nugatory crossing,

nugatory crossing

(∵) (iii) By Kidwell’s inequality x-maxdegF(D) ≤ c(D) − b(D), where b(D) is the bridge length of a diagram D. and, the equation

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Outline of the proof of Main results Prop.2 [Thistlethwaite 1988] (ii) Dalt : a reducible diagram ⇒ ϕ(D) = 0 (iii) For a non-nugatory crossing, Main lemma (ii) Dalt : a reducible diagram of nontrivial knot ⇒ π(D) = 0

, for non-nugatory

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Outline of the proof of Main lemma (iii) At the relevant crossing P(D0) = z(P(D5) + P(D7)), by Morton’s inequality.

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Topic related to ϕ(K) The chromatic invariant κ(G) is the graph invariant defined as follows: (1) (2) If G has a cut-edge or a loop, then κ(G) = 0 (3) For non cut-edge, Prop.3 [Thistlethwaite 1988] K : an alternating prime knot. G : a plane graph associated with a reduced diagram of K.

➔

G ⇒ ϕ(K) = κ(G).

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Topic related to ϕ(K) The chromatic invariant κ(G) is the graph invariant defined as follows: (1) (2) If G has a cut-edge or a loop, then κ(G) = 0 (3) For non cut-edge, Remind (ii) Dalt : a reducible diagram ⇒ ϕ(D) = 0 (iii) For a non-nugatory crossing,

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Topic related to π(K) gc(K) : the canonical genus of a knot K := min{g(S)|S is a Seifert surface obtained by Seifert’s algorithm}

☛ ✡ ✟ ✠

Conj.3 [Tripp 2002] K : a prime knot ⇒ gc(W(K)) = c(K). Fact z-maxdegP(W(K)) ≤ 2gc(W(K)) ≤ 2c(K).

✛ ✚ ✘ ✙

Conj.4 [Tripp 2002, Nakamura 2006, Brittenham-Jensen 2006] K : an alternating prime knot ⇒ π(K) ̸= 0, therefore gc(W(K)) = c(K).

• Cor. K : alternating prime knot s.t. c(K) ≤ 12
• r satisfying the condition of Main result 2.

⇒ ϕ(K) = π(K) ̸= 0, therefore gc(W(K)) = c(K).

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Further research

• We can not obtain the following knot K by Main results.

ϕ(K) = 11. π(K) = ?

• We propose the following conjectures.

✓ ✒ ✏ ✑

Main Conj.1 K : an alternating prime knot ⇒ ϕ(K) = π(K).

✛ ✚ ✘ ✙

Main Conj.2 For any non-nugatory crossing,

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