Knot invariant

For example, knot genus is particularly tricky to compute, but can be effective (for instance, in distinguishing mutants).

Volume, and other hyperbolic invariants, have proven very effective, utilized in some of the extensive efforts at knot tabulation.

Along a different line of study, there is a combinatorially defined cohomology theory of knots called Khovanov homology whose Euler characteristic is the Jones polynomial.

This has recently been shown to be useful in obtaining bounds on slice genus whose earlier proofs required gauge theory.

Mikhail Khovanov and Lev Rozansky have since defined several other related cohomology theories whose Euler characteristics recover other classical invariants.

Catharina Stroppel gave a representation theoretic interpretation of Khovanov homology by categorifying quantum group invariants.

An old result in this direction is the Fáry–Milnor theorem states that if the total curvature of a knot K in