Concordance of certain 3-braids and Gauss diagrams

TL;DR

This paper investigates the properties of knots formed by certain 3-braids using Gauss diagrams, revealing conditions under which these knots are algebraically slice and connecting these findings to Lucas numbers.

Contribution

It introduces a novel application of Gauss diagram formulas to analyze 3-braid closures and establishes a new criterion for their algebraic sliceness based on the exponent.

Findings

Knots from the closure of $eta^n$ are algebraically slice iff n is odd, for n not divisible by 3.

Provides a link between braid properties and Lucas numbers.

Identifies specific algebraic conditions for 3-braid closures.

Abstract

Let $\beta:=\sigma_1\sigma_2^{-1}$ be a braid in $B_3$, where $B_3$ is the braid group on 3 strings and $\sigma_1, \sigma_2$ are the standard Artin generators. We use Gauss diagram formulas to show that for each natural number $n$ not divisible by $3$ the knot which is represented by the closure of the braid $\beta^n$ is algebraically slice if and only if $n$ is odd. As a consequence, we deduce some properties of Lucas numbers.