Month: January 2022

Knots in overtwisted contact manifolds

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Rima Chaterjee (University of Cologne)

Knots associated to overtwisted manifolds are less explored. There are two types of knots in an overtwisted manifold – loose and non-loose. Non-loose knots are knots with tight complements where as
loose knots have overtwisted complements. While we understand loose knots, non-loose knots remain a mystery. The classification and structure problems of these knots vary greatly compared to the knots
in tight manifolds. In this talk, I’ll give a brief survey followed by some interesting recent work. Especially I’ll show how satellite operations on a knot in overtwisted manifold changes the geometric
property of the knot. I will discuss under what conditions cabling operation on a non-loose knot preserves non-looseness. The ”recent part” of this talk is based on joint work  with Etnyre, Min and Mukherjee.

The Jones Polynomial from a Goeritz Matrix

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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.