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A lower bound for the genus of a knot using the Links-Gould invariant

Ben-Michael Kohli, Guillaume Tahar

Abstract

The Links-Gould invariant of links $LG^{2,1}$ is a two-variable generalization of the Alexander-Conway polynomial. Using representation theory of $U_{q}\mathfrak{gl}(2 \vert 1)$, we prove that the degree of the Links-Gould polynomial provides a lower bound on the Seifert genus of any knot, therefore improving the bound known as the Seifert inequality in the case of the Alexander invariant. One practical consequence of this new genus bound is a straightforward proof of the fact that the Kinoshita-Terasaka and Conway knots have genus greater or equal to 2.

A lower bound for the genus of a knot using the Links-Gould invariant

Abstract

The Links-Gould invariant of links is a two-variable generalization of the Alexander-Conway polynomial. Using representation theory of , we prove that the degree of the Links-Gould polynomial provides a lower bound on the Seifert genus of any knot, therefore improving the bound known as the Seifert inequality in the case of the Alexander invariant. One practical consequence of this new genus bound is a straightforward proof of the fact that the Kinoshita-Terasaka and Conway knots have genus greater or equal to 2.
Paper Structure (24 sections, 21 theorems, 99 equations, 19 figures)

This paper contains 24 sections, 21 theorems, 99 equations, 19 figures.

Table of Contents

  1. Introduction
  2. Representations of Hopf superalgebra $U_{q}\mathfrak{gl}(2 \vert 1)$
  3. Generators and relations for Hopf superalgebra $U_{q}\mathfrak{gl}(2 \vert 1)$
  4. Generators
  5. $\mathbb{Z}/2\mathbb{Z}$-grading
  6. Relations
  7. A Hopf superalgebra structure on $U_{q}\mathfrak{gl}(2 \vert 1)$
  8. Coproduct
  9. Co-unit
  10. Antipode
  11. Highest weight representations on $U_{q}\mathfrak{gl}(2 \vert 1)$
  12. Bosonization of Hopf superalgebra $U_{q}\mathfrak{gl}(2 \vert 1)$
  13. Dual representations $V_{\alpha}^{\ast}$
  14. Tensor representations $V_{\alpha} \otimes V_{\alpha}^{\ast}$
  15. Change of basis
  16. ...and 9 more sections

Key Result

Theorem 1.1

Setting $span(\sum\limits_{i,j} a_{i,j} t_{0}^{i}t_{1}^{j}) = max \lbrace{ i-j~\vert~a_{i,j} \neq 0) \rbrace} - min \lbrace{ i-j~\vert~a_{i,j} \neq 0) \rbrace}$, for any knot $K$:

Figures (19)

  • Figure 1: A Seifert surface for a knot of genus $g$.
  • Figure 2: (4g-0)-tangle $B$.
  • Figure 3: Embedded Seifert surface $\Sigma$ of genus $g$ for a knot $K$.
  • Figure 4: Tangle $T$.
  • Figure 5: Tangle $T$ after isotopy.
  • ...and 14 more figures

Theorems & Definitions (56)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Theorem 2.8
  • ...and 46 more