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A new invariant of equivariant concordance and results on 2-bridge knots

Alessio Di Prisa, Giovanni Framba

TL;DR

The work addresses equivariant concordance for strongly invertible knots, focusing on $2$-bridge knots. It derives an explicit formula for the butterfly polynomial of the butterfly link and proves two independent obstructions showing that no $2$-bridge knot is equivariantly slice. A new invariant, the moth polynomial, converts butterfly data into a concordance-homomorphism and yields a practical infinite-order criterion via the Conway polynomial. Consequently, every $2$-bridge knot has infinite order in the directed equivariant concordance group, regardless of the strong inversion chosen, sharpening our understanding of equivariant sliceness and expanding obstruction tools.

Abstract

We study the equivariant concordance classes of two-bridge knots, providing an easy formula to compute their butterfly polynomial, and we give two different proofs that no two-bridge knot is equivariantly slice. Finally, we introduce a new invariant of equivariant concordance for strongly invertible knots. Using this invariant as an obstruction we strengthen the result on two-bridge knots, proving that their equivariant concordance order is always infinite.

A new invariant of equivariant concordance and results on 2-bridge knots

TL;DR

The work addresses equivariant concordance for strongly invertible knots, focusing on -bridge knots. It derives an explicit formula for the butterfly polynomial of the butterfly link and proves two independent obstructions showing that no -bridge knot is equivariantly slice. A new invariant, the moth polynomial, converts butterfly data into a concordance-homomorphism and yields a practical infinite-order criterion via the Conway polynomial. Consequently, every -bridge knot has infinite order in the directed equivariant concordance group, regardless of the strong inversion chosen, sharpening our understanding of equivariant sliceness and expanding obstruction tools.

Abstract

We study the equivariant concordance classes of two-bridge knots, providing an easy formula to compute their butterfly polynomial, and we give two different proofs that no two-bridge knot is equivariantly slice. Finally, we introduce a new invariant of equivariant concordance for strongly invertible knots. Using this invariant as an obstruction we strengthen the result on two-bridge knots, proving that their equivariant concordance order is always infinite.
Paper Structure (10 sections, 11 theorems, 18 equations, 8 figures)

This paper contains 10 sections, 11 theorems, 18 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Directed strongly invertible knots
  4. Butterfly links
  5. Strong inversions on two-bridge knots
  6. Eta-function
  7. No two-bridge knot is equivariantly slice
  8. Linking number proof
  9. Nullity proof
  10. A new invariant of equivariant concordance

Key Result

Theorem 1

Let $K$ be a directed strongly invertible knot and let $\widehat{L_b}(K)$ be its butterfly link endowed with the opposite semi-orientation. If the Conway polynomial of $\widehat{L_b}(K)$ is non-zero then $K$ is not equivariantly slice and has infinite order in $\widetilde{\mathcal{C}}$.

Figures (8)

  • Figure 1: $I_1(\alpha_1,\dots,\alpha_n;c_1\dots,c_n)$.
  • Figure 3: Strong inversion on a $2$-bridge knot and construction of the butterfly link.
  • Figure 4: Flypes on the butterfly link.
  • Figure 5: Fundamental domain with labelled oriented arcs.
  • Figure 6:
  • ...and 3 more figures

Theorems & Definitions (32)

  • Theorem
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Proposition 2.6
  • proof
  • Corollary 2.7
  • Remark 3.1
  • ...and 22 more