turkmath.org

Virtual knot theory is an extension of classical knot theory to stabilized embeddings of circles into thickened orientable surfaces of genus possibly greater than zero. Classical knot theory is the case of genus zero. There is a diagrammatic theory for studying virtual knots and links, and this diagrammatic theory lends itself to the construction of numerous new invariants of virtual knots as well as extensions of known invariants. The Jones polynomial extends naturally to virtual knots and gives rise to many significant problems in this domain. For example, there are infinitely many virtual knots with unit Jones polynomial. It is an open problem whether there are any classical knots (nontrivial ones) that have unit Jones polynomial. Thus we hope to shed light on the nature of the classical problem by studying its virtual counterpart. The talk will be self-contained, with many mathematical examples.