Concordance of certain 3-braids and Gauss diagrams

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.