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An algorithm for Seifert surfaces in 3-manifolds via surgery presentations

Geunyoung Kim

Abstract

The classical Seifert algorithm provides an explicit construction of a Seifert surface for any link in $S^3$. Alegria and Menasco extended this construction to integral homology $3$-spheres using Heegaard splittings. In this paper, we extend the Seifert algorithm to null-homologous links in arbitrary $3$-manifolds via surgery on framed links in $S^3$.

An algorithm for Seifert surfaces in 3-manifolds via surgery presentations

Abstract

The classical Seifert algorithm provides an explicit construction of a Seifert surface for any link in . Alegria and Menasco extended this construction to integral homology -spheres using Heegaard splittings. In this paper, we extend the Seifert algorithm to null-homologous links in arbitrary -manifolds via surgery on framed links in .
Paper Structure (3 sections, 8 theorems, 39 equations, 1 figure)

This paper contains 3 sections, 8 theorems, 39 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main theorem

Key Result

Theorem 1.1

Let $Y=S^3(L,\phi)$ be a $3$-manifold obtained by surgery on a framed link $(L,\phi)$ in $S^3$. Assume that $K\subset S^3\setminus\operatorname{int}(\nu(L))$ is an oriented null-homologous link in $Y$. Then there exists an explicit algorithm which isotopes $K$ in $Y$ to a link $K'$ that bounds a Sei

Figures (1)

  • Figure 1:

Theorems & Definitions (21)

  • Theorem 1.1
  • Corollary 1.1
  • Corollary 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Remark 2.6
  • ...and 11 more