# nLab knot sum

### Context

#### Knot theory

knot theory

Examples/classes:

Types

knot invariants

Related concepts:

category: knot theory

# Contents

## Definition

First, the connected sum $K_1 # K_2$ of boxed knots $K_1$ and $K_2$ is obtained by joining the boxes containing them.

From this one obtains a notion of connected sum for ordinary (oriented) knots by observing that:

###### Proposition

There is a canonical bijection between the equivalence classes of boxed knots and the isotopy classes of oriented knots.

This bijection gives the connected sum of pairs of ordinary knots.

## Properties

###### Theorem

The connected sum operation is associative and commutative:

• $K_1 # K_2 = K_2 # K_1$,

• $( K_1 # K_2 ) # K_3 = K_1 # ( K_2 # K_3 )$.

There are no inverse elements under the connected sum operation, i.e., $K # K' = \circ \implies K = K' = \circ$, where $\circ$ is the unknot.

## References

Discussion in relation to Vassiliev knot invariants:

Last revised on June 28, 2024 at 12:42:20. See the history of this page for a list of all contributions to it.