Link Invariants
Examples
Related concepts
A knot invariant is map from isotopy equivalence classes of knots to any kind of structure you could imagine. These are helpful because it is much easier to check that the structures one maps to (numbers, groups, etc.) are different than it is to check that knots are different.
Many of these extend to link invariants or have variants that depend on the knot being oriented.
Discussion of knot invariants in terms of BPS states in string theory includes
Edward Witten, Fivebranes and Knots (arXiv:1101.3216)
Sergei Gukov, Marko Stošić, Homological algebra of knots and BPS states (arXiv:1112.0030)
Ross Elliot, Sergei Gukov, Exceptional knot homology (arXiv:1505.01635)