Surface Links, Polynomial Coefficients, and Volume

Andrew Will (Brooklyn College)

Abstract: Invariants are powerful tools for distinguishing knots and links which come in many flavors and varieties. Among these are the colored Jones polynomials, the subject of the famed Volume Conjecture. This surprising open problem purports to bridge the gap between diagrammatic and geometric invariants by asserting that a hyperbolic link’s colored Jones polynomials determine the volume of its complement. In this talk, we will give an overview of the “Volume-ish” Theorem of Dasbach—Lin, a weaker result under the umbrella of the Volume Conjecture. We will also discuss the theory of Jones-like polynomials for links in thickened surfaces as well as an extension of the theorem to this setting.