Goeritz matrices

The Jones Polynomial from a Goeritz Matrix

| chrisman.76

Joe Boninger (CUNY)

The Jones polynomial holds a central place in knot theory, but its topological meaning is not well understood—it remains an open problem, posed by Atiyah, to give a three-dimensional interpretation of the polynomial. In this talk, we’ll share an original construction of the Jones polynomial from a Goeritz matrix, a combinatorial object with topological significance. In the process we extend the Kauffman bracket to any symmetric, integer matrix, with applications to links in thickened surfaces. Matroid theory plays a role.

The Gordon-Litherland pairing for knots and links in thickened surfaces

| chrisman.76

Hans Boden (McMaster University)

We introduce the Gordon-Litherland pairing for knots and links in thickened surfaces that bound unoriented spanning surfaces. Using the GL pairing, we define signature and determinant invariants for such links. We relate the invariants to those derived from the Tait graph and Goeritz matrices. These invariants depend only on the $S^*$ equivalence class of the spanning surface, and the determinants give a simple criterion to check if the knot or link is minimal genus. This is joint work with M. Chrisman and H. Karimi. In further joint work with H. Karimi, we apply the GL pairing to give a topological characterization of alternating links in thickened surfaces, extending the results of J. Greene and J. Howie. 

Slides for talk:

GL-pairing-talk