Ropelength, crossing number and finite type invariants of links

TL;DR

This paper investigates the relationship between geometric complexity measures like ropelength and crossing number of links, providing new lower bounds based on finite type invariants and extending previous linking number bounds.

Contribution

It offers new estimates for embedding thickness of links in terms of Milnor linking numbers, generalizing known bounds and analyzing finite type invariants and their impact on knot complexity.

Findings

Derived lower bounds for embedding thickness using Milnor linking numbers.

Identified cases where these bounds outperform knot-genus estimates.

Collected facts about finite type invariants and their relation to knot complexity.

Abstract

Ropelength and embedding thickness are related measures of geometric complexity of classical knots and links in Euclidean space. In their recent work, Freedman and Krushkal posed a question regarding lower bounds for embedding thickness of $n$-component links in terms of the Milnor linking numbers. The main goal of the current paper is to provide such estimates and thus generalizing the known linking number bound. In the process, we collect several facts about finite type invariants and ropelength/crossing number of knots. We give examples of families of knots, where such estimates behave better than the well-known knot-genus estimate.