Session S28 - Knots, Surfaces, 3-manifolds

Wednesday, July 14, 17:20 ~ 17:50 UTC-3

The Gordon-Litherland pairing for links in thickened surfaces

Hans Boden

McMaster University, Canada   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

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 and relate them to invariants 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 has minimal genus. The GL pairing is isometric to the relative intersection pairing on a 4-manifold obtained as the 2-fold cover along the surface. Time permitting, we will explain how to use the GL pairing to give a topological characterization of alternating links in thickened surfaces, extending the results of Josh Greene and Josh Howie.

Joint work with Micah Chrisman (Ohio State University, Marion) and Homayun Karimi (McMaster University).

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