Concordance of certain 3-braids and Gauss diagrams

Let beta := sigma(1)sigma(-1)(2) 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. (C) 2016 Elsevier B.V. All rights reserved.