---
tag: JKA25KDPQP
kind: concept
name: Alexander polynomial
msc: [57K14, 57K10]
specializes: [GZQ5ZE3X6G]
sources:
  - url: https://en.wikipedia.org/wiki/Alexander_polynomial
same_as:
  wikidata: Q1634206
  wikipedia: Alexander polynomial
---

## Summary

The Alexander polynomial is a [[knot invariant|GZQ5ZE3X6G]] that assigns to each [[knot|QYEAZTYGPT]] a Laurent polynomial with integer coefficients.
It was the first polynomial knot invariant, found by James Waddell Alexander II in the 1920s.
It can be computed from a Seifert matrix, from a knot diagram, or from the homology of the infinite cyclic cover of the knot complement.
It detects many knots but not all; some nontrivial knots have Alexander polynomial $1$.

## Explanations

### L1 human

The Alexander polynomial turns a knot into a polynomial such as $t - 1 + t^{-1}$.
If two knots have different polynomials, they are different knots.
The [[unknot|1TQ06R4Q66]] has polynomial $1$, while the [[trefoil knot|VYTYSVHN4A]] has $t - 1 + t^{-1}$, so the trefoil is really knotted.
The polynomial records how the loops on a surface bounded by the knot wind around each other.