---
tag: 1C28287S75
kind: concept
name: Knot concordance
msc: [57K10, 57K40]
sources:
  - url: https://en.wikipedia.org/wiki/Slice_knot
---

## Summary

Two oriented [[knots|QYEAZTYGPT]] are concordant if they cobound a smoothly embedded annulus in $S^3 \times [0, 1]$.
Concordance is an equivalence relation, coarser than isotopy.
The concordance classes form an abelian group under [[knot sum|AGZSZW7GYE]], the knot concordance group.
Its zero element is the class of [[slice knots|P68G28E4AD]].

## Explanations

### L1 human

Think of time as a fourth direction, and let a knot move through time.
During the motion it may change shape in ways that are not allowed in ordinary space, as long as the trace it sweeps out is a smooth tube without self-crossings.
Two knots are concordant if one can be turned into the other in this way.
Under this coarser notion of sameness, every knot has an inverse: its reversed mirror image.