Knot invariant

Remarks

Equivalently, a knot invariant is a map on the set of equivalence classes of knots. The values may be numbers, polynomials, groups up to isomorphism, or homology theories up to isomorphism. Example: the crossing number is a knot invariant with values in $\N$, because it is a minimum taken over the diagrams of all knots equivalent to $K$. Non-example: the number of crossings of one chosen diagram of $K$ is not an invariant of $K$. Indeed, a twist in a diagram of the unknot adds a crossing without changing the knot. An invariant is complete if $f(K) = f(K')$ implies that $K$ and $K'$ are equivalent. The same notion is used for links.