Examples/classes:
Types
Related concepts:
transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
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:
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:
$K_1 # K_2 = K_2 # K_1$,
$( K_1 # K_2 ) # K_3 = K_1 # ( K_2 # K_3 )$.
There are no inverse elements under the connected sum operation, i.e., $K # K' = \circ \implies K = K' = \circ$, where $\circ$ is the unknot.
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.