Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A partial classification of 3-dimensional clasp number two, genus two fibered knots

Tetsuya Ito

TL;DR

This work studies genus two fibered knots with clasp number two by focusing on knots that admit a clasp disk of type II. It shows that such knots can be constructed from a base genus-one fibered knot or a genus-zero 3-component fibered link via two simultaneous Hopf plumbings, reducing the problem to the classification of genus-zero 3-component fibered links and their attaching arcs. The main result is an explicit catalog of all genus two fibered knots with $cl(K)=2$ (up to mirror image) arising from type II clasp disks: non-prime sums, certain 2-bridge knots, a family of Montesinos knots, and 12 exceptional examples; the work also provides Conway and HOMFLY polynomial constraints that distinguish type II from type X and connects these invariants to the clasp data. The approach integrates Hopf plumbing, open-book decompositions, and rational-tangle/Montesinos techniques to deliver a concrete local-to-global classification with potential implications for understanding how knot invariants reflect clasp-number constraints.

Abstract

The 3-dimensional clasp number $cl(K)$ of a knot $K$ is the minimum number of clasp singularities of clasp disk, a singular immersed disk bounding $K$ whose singular set consists of only clasp singularities. We give a classification of clasp number two, genus two fibered knots under the assumption that they admit a clasp disk of certain type which we call of type II.

A partial classification of 3-dimensional clasp number two, genus two fibered knots

TL;DR

This work studies genus two fibered knots with clasp number two by focusing on knots that admit a clasp disk of type II. It shows that such knots can be constructed from a base genus-one fibered knot or a genus-zero 3-component fibered link via two simultaneous Hopf plumbings, reducing the problem to the classification of genus-zero 3-component fibered links and their attaching arcs. The main result is an explicit catalog of all genus two fibered knots with (up to mirror image) arising from type II clasp disks: non-prime sums, certain 2-bridge knots, a family of Montesinos knots, and 12 exceptional examples; the work also provides Conway and HOMFLY polynomial constraints that distinguish type II from type X and connects these invariants to the clasp data. The approach integrates Hopf plumbing, open-book decompositions, and rational-tangle/Montesinos techniques to deliver a concrete local-to-global classification with potential implications for understanding how knot invariants reflect clasp-number constraints.

Abstract

The 3-dimensional clasp number of a knot is the minimum number of clasp singularities of clasp disk, a singular immersed disk bounding whose singular set consists of only clasp singularities. We give a classification of clasp number two, genus two fibered knots under the assumption that they admit a clasp disk of certain type which we call of type II.
Paper Structure (7 sections, 10 theorems, 34 equations, 10 figures)

This paper contains 7 sections, 10 theorems, 34 equations, 10 figures.

Table of Contents

  1. Introduction
  2. A quick review of Conway and HOMFLY polynomial of clasp number two knots
  3. Conway polynomial
  4. The (zero-th) coefficient of the HOMFLY polynomial
  5. Classification
  6. Classification of three component, genus zero fibered links in $S^{3}$
  7. Rational tangle and Montesinos knot

Key Result

Theorem 1.1

Let $K$ be a genus two fibered knot with $cl(K)=2$. If $K$ admits a clasp disk of type II, then up to taking mirror image, $K$ is one the following knot.

Figures (10)

  • Figure 1: (a) positive and negative clasp singularity. (b) The Seifert surface from clasp disk. (c) resolution of clasp singularity. (d) smoothing of clasp singularity
  • Figure 2: 12 exceptional knots $K^{\sf ex}_{i;\varepsilon_1,\varepsilon_2}$
  • Figure 3: Skein tree: the knots $K_{u,\gamma_2}$, $K_{o,\gamma_2}$,$K_{u,u}$,$K_{u,o}$,$K_{o,u}$, $K_{o,o}$
  • Figure 4: (a) Clasp disk of type X. (b) Clasp disk of type II. A box represents some tangle.
  • Figure 5: (i) Basis for type X case. (ii) Basis for type II case. A key feature is that the component $K_1$ is equal to the component of $K_{o,u}$ and the component $K_2$ is equal to $K_2$.
  • ...and 5 more figures

Theorems & Definitions (21)

  • Theorem 1.1: Classification of genus two, clasp number two fibered knots of type II
  • Corollary 1.2
  • proof
  • Definition 2.1
  • Theorem 2.2
  • Corollary 2.3
  • proof
  • Remark 2.4
  • Remark 2.5
  • Theorem 2.6
  • ...and 11 more