Summary: Mutations of Links in Genus 2 Handlebodies
by
D.Cooper* and W.B.R.Lickorish**
Abstract A short proof is given to show that a link in the 3­sphere and any link related to it
by genus 2 mutation have the same Alexander polynomial. This verifies a deduction from the solution to the
Melvin­Morton conjecture. The proof here extends to show that the link signatures are likewise the same
and that these results extend to links in a homology 3­sphere.
1. Introduction A remarkable correlation between the Jones polynomials of a classical
knot and its Alexander polynomial was established by D.Bar­Natan and S.Garoufalidis
in [1], where they gave a proof of a conjecture of P.M.Melvin and H.R.Morton. The
background to the conjecture was established in [4] and heuristic evidence in its favour
was given by L.Rozansky [7]. The conjecture asserted that, with a specific formula, it
was possible to calculate each coefficient in the Alexander polynomial of a knot from
knowledge of all the coefficients in all the coloured Jones polynomials of the knot (with
zero framing). These coloured Jones polynomials can be regarded as the generalisations of
the Jones invariant to framed knots equipped with irreducible representations of SU(2) or,
interpretted by means of the Kauffman bracket skein, as the invariant of the zero­framed
knot decorated by a Chebyshev polynomial in the skein generator of the solid torus. The
proof given in [1] is an intricate and skillful deployment of Vassiliev invariants analysed by
means of arguments with chord diagrams and weight systems. Indeed, it was in the world

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

Collections: Mathematics