---
tag: K866ND3WF7
kind: concept
name: Seifert surface
msc: [57K10]
sources:
  - url: https://en.wikipedia.org/wiki/Seifert_surface
same_as:
  wikidata: Q1554293
  wikipedia: Seifert surface
---

## Summary

A Seifert surface of a [[knot|QYEAZTYGPT]] or [[link|1A5WRCNTM3]] is a compact, connected, orientable surface in space whose boundary is the knot or link.
Every oriented link has one, by an algorithm of Herbert Seifert from 1934.
The least [[genus|NJ787GZS1N]] of a Seifert surface of a knot is the genus of the knot, a [[knot invariant|GZQ5ZE3X6G]].
Seifert surfaces are also used to define the Alexander polynomial and the signature of a knot.

## Explanations

### L1 human

Dip a knotted wire loop into soap solution: the soap film that forms is a surface whose edge is the knot.
A Seifert surface is such a surface with two distinct sides, so that it can be painted in two colours.
For a round circle, a flat disk works.
For more complicated knots, the surface needs handles, and the least number of handles needed is the genus of the knot.