August 7, 2025

On the symmetric braid index of ribbon knots

Key Points

MAIN FINDING: The symmetric braid index $b_s(K)$ provides insights into ribbon knots' structure.
KEY EVIDENCE: Knots exist where $b_s(K)$ exceeds the traditional braid index, highlighting knot complexity.
APPROACH: The research defines $b_s(K)$ and utilizes Khovanov-homological techniques for analysis.
SIGNIFICANCE: These findings enhance our understanding of knot theory and its applications in topology.

Abstract

Abstract We define the symmetric braid index bₛ (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 bₛ (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 bₛ (K) for prime ribbon knots with at most 11 crossings.

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Cite This Study

Brejevs et al. (Tue,) studied this question.

synapsesocial.com/papers/689521d79f4f1c896c427907 https://doi.org/https://doi.org/10.1017/s0305004125101424

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