The ropelength conjecture of alternating knots

TL;DR

This paper proves the long-standing conjecture that the ropelength of any alternating knot is at least proportional to its crossing number, establishing a fundamental geometric relationship in knot theory.

Contribution

The paper confirms the conjecture by demonstrating a proportional lower bound for ropelength in terms of crossing number for all alternating knots.

Findings

Existence of a constant $b_0>0$ such that $R(K) _0Cr(K)$ for all alternating knots

Proof of the proportional lower bound for ropelength in relation to crossing number

Validation of the long-standing conjecture in knot theory

Abstract

A long standing conjecture states that the ropelength of any alternating knot is at least proportional to its crossing number. In this paper we prove that this conjecture is true. That is, there exists a constant $b_0>0$ such that $R(K)\ge b_0Cr(K)$ for any alternating knot $K$, where $R(K)$ is the ropelength of $K$ and $Cr(K)$ is the crossing number of $K$. In this paper, we prove that this conjecture is true.