On the symmetric braid index of ribbon knots

TL;DR

This paper introduces the symmetric braid index for ribbon knots, characterizes knots with small symmetric braid index using Khovanov homology, and explores its implications for knot properties and bounds.

Contribution

It defines the symmetric braid index for ribbon knots, provides a Khovanov homological characterization for small indices, and investigates its relation to other knot invariants.

Findings

Existence of knots with symmetric braid index greater than braid index

Chiral slice knots with determinant one have braid index at least four

Bounds for symmetric braid index of prime ribbon knots with up to 11 crossings

Abstract

We define the symmetric braid index $b_s(K)$ of a ribbon knot $K$ to be the smallest index of a braid whose closure yields a symmetric union diagram of $K$, and derive a Khovanov-homological characterisation of knots with $b_s(K)$ at most three. As applications, we show that there exist knots whose symmetric braid index is strictly greater than the braid index, and deduce that every chiral slice knot with determinant one has braid index at least four. We also calculate bounds for $b_s(K)$ for prime ribbon knots with at most 11 crossings.