Archirality of Alternating Knots

Quách Cẩm Vân Hongler

Abstract: An achiral knot in the 3-space is a knot which is equivalent to its mirror image. We give a review on the problem whether the achirality of an alternating knot in the 3-space can be detected from an alternating projection.

1) By performing the Murasugi decomposition on Seifert surfaces, a nice topological criteria for achiral alternating knots and a nice property on their Conway polynomials are deduced. Therefore in most cases, we can deduce whether an alternating knot is not achiral.

2) Related to the problem of the visibility of achirality, L. Kauffman conjectured about the checkboard surfaces of an alternating knot the following fact: an achiral alternating knot has always an alternating projection such that the associated checkboard graphs are dual. This conjecture is true for negative achiral alternating knots. For positive alternating achiral knots, it is proven the conjecture can be false only for the case with order of achirality 4.

Time: 13:00, 09/05/2016, F207.