Knot sum

Builds on Knot

Definition

Let $K_1$ and $K_2$ be oriented tame knots in $S^3$ . Choose disjoint closed $3$ -balls $B_1, B_2 \subset S^3$ with $K_i \subset B_i$ . Choose a band $b \cong [0,1] \times [0,1]$ in $S^3$ that meets $B_1 \cup B_2$ only in $\set{0, 1} \times [0,1]$ . Require $\set{i-1} \times [0,1] \subset K_i$ for $i = 1, 2$ . Choose the band so that the orientations of $K_1$ and $K_2$ agree with an orientation of the knot below. The knot sum $K_1 \# K_2$ is the oriented knot $\big( (K_1 \cup K_2) \setminus (\set{0,1} \times (0,1)) \big) \cup \big( [0,1] \times \set{0, 1} \big).$

Remarks

The isotopy class of $K_1 \# K_2$ does not depend on the balls or the band. Example: if $K_2$ is the unknot, the band slides into a trivial arc and $K_1 \# K_2$ is isotopic to $K_1$ . Non-example: if the band is allowed to enter $B_1$ away from $K_1$ , it may wind around $K_1$ . Gluing along such a band can give a knot different from $K_1 \# K_2$ , so the condition on the band matters. The sum is commutative and associative on isotopy classes of oriented knots. The genus of a Seifert surface is additive, $g(K_1 \# K_2) = g(K_1) + g(K_2)$ , so a nontrivial knot has no inverse. In a diagram, the sum is drawn by placing the two diagrams side by side and replacing two facing arcs by two parallel arcs.

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Knot concept
Knot sum definition

Tag	`4ACJ5TPHJX`
URI	`m2://4ACJ5TPHJX`
Kind	definition
Writing	W0 stub
MSC	57K10
Defines	Knot sum
Uses	Knot
Changed	2 November 2026

Knot sum

Definition

Remarks

Reading path

Further reading

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