モンテシノス結び目の 境界スロープの評価について

日本数学会2007_年度年会 (2007.3.30)

モンテシノス結び目の

境界スロープの評価について

市原 一裕

_{大阪産業大学教養部}

水嶋 滋

東京工業大学情報理工学研究科

との共同研究

Boundary slope :

Let E(K) be the exterior of a knot K. Definition (Essential surface)

A surface F embedded in E(K) is called essential if it is incompressible and ∂-incompressible.

Definition (Boundary slope)

The slope on ∂E(K) determined by ∂F is called the ∂-slope of F .

Recall that : using meridian-longitude system,

slopes are represented by rational numbers with 1/0.

Montesinos knot: means a knot in the 3-sphere S³ obtained by connecting rational tangles in line.

Assume that:

· the number of tangles ≥ 3

· all tangles are non-integral

Remark :

In the following, we assume that all surfaces are orientable just for simplifying the statements.

In fact, we have obtained general results including non-orientable case.

(1) Denominator of ∂-slope

Let p/q be a ∂-slope of an essential surface of genus g for a knot K.

Facts.

q ≤ g holds if K is a composite knot, (Torisu)

or, an alternating knot (Menasco-Thistlethwaite).

Theorem 1

If K is a Montesinos knot, then







q ≤ g + 1 (g = 0, 1) q ≤ 2g − 1 (g ≥ 2)

(2) Difference between ∂-slopes

Let r₁, r₂ be ∂-slopes of essential surfaces of genus g₁, g₂ for K.

Theorem 2

If K is a Montesinos knot, then |r₁−r₂| ≤ 4 (g₁+g₂) Let r_M, r_m be the maximal, minimal ∂-slopes for K.

Theorem 3

There exist essential surfaces F_M, F_m with ∂-slopes r_M, r_m such that r_M − r_m ≥ 2

(−χ(F₁)

s(F₁) + ^−χ(F2)

s(F₂)

)

.

(3) Diameter of boundary slope set

Definition (Boundary slope set)

The set of ∂-slopes for K is called the boundary slope set B_K.

Definition (Diameter of B_K)

The diameter Diam(K) of BK is defined as max{|r − r⁰| | r, r⁰ ∈ B_K}

Fact (Culler-Shalen)

If K is a non-trivial knot, then Diam(K) ≥ 2.

Let Cr(K) denote the minimal crossing number of K. Fact

If K is an alternating knot, then 2 Cr(K) ≤ Diam(K).

Theorem 4

If K is a Montesinos knot,

then 2 Cr(K) − 6 ≤ Diam(K) ≤ 2 Cr(K).

Conjecture

In general, Diam(K) ≤ 2 Cr(K). In particular, if K is alternating, then 2 Cr(K) = Diam(K).

(4) Absolute value of ∂-slope: NEW

Let r be the ∂-slope of an essential surface, and let Cr₊(D₀), Cr₋(D₀) be the number of positive,negative crossings of a minimal diagram D₀ for K.

“Theorem” 5

If K is a Montesinos knot,

then −2Cr₋(D₀) ≤ r ≤ 2Cr₊(D₀).

Let wr(D₀) denote the writhe of the diagram D₀. Corollary

If K is a Montesinos knot, then |r| ≤ Cr(D₀)+|wr(D₀)|.

Key of Proofs :

Algorithm by Hatcher and Oertel

can enumerate all boundary slopes for a given Mon- tesinos knot.

c.f.

Dunfield’s software

implements their algorithm.