Chord Index type invariants of virtual knots

Zhiyun Cheng (Beijing Normal University)

As an extension of classical knot theory, virtual knot theory studies the embeddings of one sphere in thickened surfaces up to stable equivalence. Roughly speaking, there are two kinds of virtual knot invariants, the first kind comes from knot invariants of classical knots but the second kind usually vanishes on classical knots. Most of the second kind of virtual knot invariants are defined by using the chord parity or chord index. In this talk, I will report some recent progress on virtual knot invariants derived from various chord indices.

Minimal crossing number implies minimal supporting genus

Will Rushworth (McMaster University)

We prove that a minimal crossing virtual link diagram is a minimal genus diagram. That is, the genus of its Carter surface realizes the supporting genus of the virtual link represented by the diagram. The result is obtained by introducing a new parity theory for virtual links. This answers a basic question in virtual knot theory, and recovers the corresponding result of Manturov in the case of virtual knots.
Joint work with Hans Boden.