Infinite families of prime knots with alt(K) = 1 and their Alexander polynomials

Abstract

In this paper, we construct, by using the Alexander polynomial, infinite families of nonalternating prime knots, which have alternation number equal to one. More specifically these knots after one crossing change yield a 2-bridge knot or the trivial knot. In particular, we display two infinite families of nonalternating knots and their Alexander polynomials. Moreover, we give formulae to obtain the Conway and Alexander polynomials of oriented 3-tangles and the links formed from their closure with a specific orientation. In particular, we propose a construction to form families of links for which their Alexander polynomials can be obtained by nonrecursive formulae.