From Bulletin of the AMS
Categorical lifting of the Jones polynomial: a survey
by
Mikhail Khovanov and
Robert Lipshitz
This paper is dedicated to the memory of Vaughan Jones, whose insights have illuminated so many beautiful mathematical paths.
Abstract
This is a brief review of the categorification of the Jones polynomial and its significance and ramifications in geometry, algebra, and low-dimensional topology.
1. Constructions of the Jones polynomial
The spectacular discovery by Vaughan Jones Reference [76] [78] of the Jones polynomial of links has led to many follow-up developments in mathematics. In this note we will survey one of these developments, the discovery of a combinatorially defined homology theory of links, functorial under link cobordisms in 4-space, and its connections to algebraic geometry, symplectic geometry, gauge theory, representation theory, and stable homotopy theory.
The Jones polynomial $J(L)$ of an oriented link $L$ in $\mathbb{R}^3$ is determined uniquely by the skein relation …
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