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Alternating links have at most polynomially many Seifert surfaces of fixed genus

Joel Hass, Abigail Thompson, Anastasiia Tsvietkova

Abstract

Let $L$ be a non-split prime alternating link with $n>0$ crossings. We show that for each fixed $g$, the number of genus-$g$ Seifert surfaces for $L$ is bounded by an explicitly given polynomial in $n$. The result also holds for all spanning surfaces of fixed Euler characteristic. Previously known bounds were exponential.

Alternating links have at most polynomially many Seifert surfaces of fixed genus

Abstract

Let be a non-split prime alternating link with crossings. We show that for each fixed , the number of genus- Seifert surfaces for is bounded by an explicitly given polynomial in . The result also holds for all spanning surfaces of fixed Euler characteristic. Previously known bounds were exponential.

Paper Structure

This paper contains 6 sections, 8 theorems, 1 equation, 3 figures.

Table of Contents

  1. Introduction
  2. Standard position for Seifert surfaces
  3. Decomposing a surface into regions
  4. Collections of $BBBB$ curves
  5. The number of Seifert surfaces of fixed genus
  6. Acknowledgments

Key Result

Lemma 2.1

Suppose a spanning surface $F$ has minimal complexity. Then the curves of $F\cap S_+$ and $F\cap S_-$ and the associated words in the letters $B$, $S$ have the following properties:

Figures (3)

  • Figure 1:
  • Figure 2: The link $L$ and a $BBSSS$ curve from $F\cap S_+$
  • Figure 3: A connected collection of two regions, each bounded by a $BBBB$ curve.

Theorems & Definitions (20)

  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 5.1
  • ...and 10 more