Alexander polynomial

mathematics

Written by

The Information Architects of Encyclopaedia Britannica

Last updated

July 16, 2026

Quick Summary

The Alexander polynomial is a mathematical expression used in topology to describe characteristics of knots. Developed by mathematician James Waddell Alexander II in 1928, it provides an algebraic method for distinguishing between different knots, regardless of how they are stretched or twisted.12

A knot can be visualized as a closed loop in three-dimensional space that cannot be untangled into a simple circle without cutting it. The Alexander polynomial helps mathematicians determine if two knot diagrams represent the same knot or different ones. For example, a figure-eight knot has the Alexander polynomial (t^2 - 3t + 1). This tool was a significant advancement in knot theory, offering a way to classify these complex geometric structures.21

Summary created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

3 Britannica Sources

Ask Britannica AI about this topic

Britannica doesn't have a full article on this topic, so we've curated a list of our relevant coverage.

James W. Alexander II
…polynomial, now known as the Alexander polynomial, for distinguishing various knots regardless of how they are stretched or twisted. This was an important first step in providing an algebraic way of distinguishing knots (and therefore three-dimensional manifolds).
Read More
…polynomial, now known as the Alexander polynomial, for distinguishing various knots regardless of how they are stretched or twisted. This was an important first step in providing an algebraic way of distinguishing knots (and therefore three-dimensional manifolds).
Read More

Vaughan Jones
…suspected that these were essentially Alexander polynomials (named after the work of the American mathematician James W. Alexander in 1928), but this turned out not to be the case. For any topological displacement (without cutting the loop), the associated Alexander polynomial is unchanged, or invariant. Both Alexander’s polynomials and the…
Read More
…suspected that these were essentially Alexander polynomials (named after the work of the American mathematician James W. Alexander in 1928), but this turned out not to be the case. For any topological displacement (without cutting the loop), the associated Alexander polynomial is unchanged, or invariant. Both Alexander’s polynomials and the…
Read More