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Alexander polynomial, Dijkgraaf-Witten invariant, and Seifert fibred surgery

Haimiao Chen

TL;DR

The paper develops a bridge between Dijkgraaf-Witten invariants of semidirect product gauge groups and the Alexander polynomial to constrain Dehn surgeries that yield small Seifert fibred 3-manifolds. By computing DW invariants in two ways—via metabelian representations tied to $\Delta_K$ and via a general Seifert-manifold DW formula—the author derives arithmetic constraints on the Seifert multiplicities $a_j$ and their pairwise gcds $e_j$, expressed through cyclotomic data $\Omega_\cdot$ and $\widetilde{\Omega}_\cdot$. The results re-derive and strengthen Kadokami’s bounds, providing a practical framework to infer properties of $\Delta_K$ from possible good surgeries and to bound the surgery coefficient $k$ using $\Delta_K$. Overall, the work ties 3-manifold topology to finite-group representations and cyclotomic invariants, enabling algorithmic checks for feasible Seifert fibred surgeries from the Alexander polynomial.

Abstract

We apply Dijkgraaf-Witten invariant over an semiproduct of abelian groups to show that, if the $k/\ell$-surgery along a knot $K$ results in a small Seifert 3-manifold with multiplicities $a_1,a_2,a_3$, then many constraints on $k,a_1,a_2,a_3$ can be read off from the Alexander polynomial of $K$.

Alexander polynomial, Dijkgraaf-Witten invariant, and Seifert fibred surgery

TL;DR

The paper develops a bridge between Dijkgraaf-Witten invariants of semidirect product gauge groups and the Alexander polynomial to constrain Dehn surgeries that yield small Seifert fibred 3-manifolds. By computing DW invariants in two ways—via metabelian representations tied to and via a general Seifert-manifold DW formula—the author derives arithmetic constraints on the Seifert multiplicities and their pairwise gcds , expressed through cyclotomic data and . The results re-derive and strengthen Kadokami’s bounds, providing a practical framework to infer properties of from possible good surgeries and to bound the surgery coefficient using . Overall, the work ties 3-manifold topology to finite-group representations and cyclotomic invariants, enabling algorithmic checks for feasible Seifert fibred surgeries from the Alexander polynomial.

Abstract

We apply Dijkgraaf-Witten invariant over an semiproduct of abelian groups to show that, if the -surgery along a knot results in a small Seifert 3-manifold with multiplicities , then many constraints on can be read off from the Alexander polynomial of .

Paper Structure

This paper contains 6 sections, 12 theorems, 51 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Dijkgraaf-Witten invariant over a semidirect product and Alexander polynomial
  3. The Dijkgraaf-Witten invariant over $\mathbb{Z}_{p^m}^h\rtimes_f\mathbb{Z}_n$ of a small Seifert 3-manifold
  4. Proof of Theorem \ref{['thm:main']}
  5. Applications
  6. Proof of Theorem \ref{['thm:DW-formula']}

Key Result

Theorem 1.1

Suppose $k\ge 2$, and $K(k/\ell)$ is small Seifert fibred 3-manifold with multiplicities $a_1,a_2,a_3\ge 2$. Let $e_1=(a_2,a_3), e_2=(a_1,a_3), e_3=(a_1,a_2)$.

Figures (1)

  • Figure 1: (a) a positive crossing; (b) a negative crossing.

Theorems & Definitions (27)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Corollary 3.3
  • Lemma 4.1
  • proof
  • Remark 4.2
  • Lemma 4.3
  • ...and 17 more