State Surfaces of Links

TL;DR

This paper surveys the use of state surfaces, derived from link diagrams, in understanding the geometric and topological properties of links, including their relation to Jones polynomials and hyperbolic structures.

Contribution

It provides an overview of recent applications of state surfaces in link theory, highlighting their role in connecting diagrammatic, algebraic, and geometric link invariants.

Findings

State surfaces encode geometric information about link complements.

They relate Jones polynomials to topological invariants like crosscap number.

Recent applications demonstrate their utility in studying hyperbolic volume.

Abstract

State surfaces are spanning surfaces of links that are obtained from link diagrams guided by the combinatorics underlying Kauffman's construction of the Jones polynomial via state models. Geometric properties of such surfaces are often dictated by simple link diagrammatic criteria, and the surfaces themselves carry important information about geometric structures of link complements. State surfaces also provide a tool for studying relations between Jones polynomials and topological invariants, such as the crosscap number or invariants coming from geometric structures on link complements (e.g. hyperbolic volume). This article is brief survey on some of the recent applications of state surfaces.