Knot concordance

Definition

Let $K_0$ and $K_1$ be oriented tame knots in $S^3$ . They are concordant if there is a smoothly and properly embedded annulus $A \cong S^1 \times [0, 1]$ in $S^3 \times [0, 1]$ such that $\partial A = (K_0 \times \set{0}) \cup (K_1 \times \set{1})$ . The orientation of $A$ must induce the given orientation on $K_0$ and the opposite one on $K_1$ . A concordance class is an equivalence class under this relation.

Remarks

Isotopic knots are concordant, since the trace of an isotopy is a concordance. A knot is concordant to the unknot if and only if it is slice. The concordance classes form an abelian group $\mathcal{C}$ under knot sum, with the class of the unknot as zero. The inverse of $K$ is $K^*$ , the mirror image of $K$ with reversed orientation. Example: $K \# K^*$ is slice, so it is concordant to the unknot. Non-example: the trefoil knot is not concordant to the unknot, since its signature is nonzero. Signature is additive under knot sum, so the trefoil has infinite order in $\mathcal{C}$ .

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Knot concept
Knot sum concept
Slice knot concept
Knot concordance definition

Tag	`XXZTMNBYFZ`
URI	`m2://XXZTMNBYFZ`
Kind	definition
Writing	W0 stub
MSC	57K10, 57K40
Defines	Knot concordance
Uses	Knot, Slice knot, Knot sum
Changed	2 November 2026

Knot concordance

Definition

Remarks

Reading path

Further reading

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