New Upper Bounds on the Ribbonlength of Alternating Links with Bipartite Dual Graphs

TL;DR

This paper establishes new upper bounds on the ribbonlength of alternating links with bipartite dual graphs, providing exact values for some links and improving understanding of geometric invariants in knot theory.

Contribution

It proves a universal upper bound for ribbonlength of certain alternating links and computes exact ribbonlength for the Hopf link, advancing geometric knot theory.

Findings

Rib(L)  \,  \, c(L) for links with bipartite dual graphs

Exact ribbonlength of the Hopf link is 2 \, 

Improved bounds for small crossing number knots and links

Abstract

The ribbonlength of a link is a geometric invariant defined as the infimum of the ratio of the length to the width of a folded ribbon realization of the link. In this paper, we prove that if an alternating link admits an alternating diagram with a bipartite dual graph, then its ribbonlength satisfies $$ \mathrm{Rib}(L) \le \sqrt{3} \, c(L). $$ Using this result, we present improved upper bounds on the ribbonlength for several knots and links with small crossing numbers, and determines the exact ribbonlength of the Hopf link to be $2\sqrt{3}$.