Concordance of certain 3-braids and Gauss diagrams

Abstract

Let β := σ 1 σ −1 2 be a braid in B 3 , where B 3 is the braid group on 3 strings and σ 1 , σ 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 β n is algebraically slice if and only if n is odd. As a consequence, we deduce some properties of Lucas numbers.2000 Mathematics Subject Classification. 57.