nLab knot sum


Knot theory




First, the connected sum K 1#K 2 K_1 # K_2 of boxed knots K 1K_1 and K 2K_2 is obtained by joining the boxes containing them.

(from Sossinsky 2023)

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


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.



The connected sum operation is associative and commutative:

  • K 1#K 2=K 2#K 1 K_1 # K_2 = K_2 # K_1 ,

  • (K 1#K 2)#K 3=K 1#(K 2#K 3) ( 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=K=K=K # K' = \circ \implies K = K' = \circ , where \circ is the unknot.


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.