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From Bulletin of the AMS

Categorical lifting of the Jones polynomial: a survey

The complex associated to a tangle. Image of three dotted-line circles each with two solid lines inside in different shapes. Letters across bottom are t a b s.

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.


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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