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 often much easier to check that the structures one maps to (numbers, groups, etc.) are different than it is to check that knots are different. To define a knot invariant, it suffices to define its value on knot diagrams and check that this value is preserved under the Reidemeister moves (possibly with the exception of the first Reidemeister move, in the case of an invariant of framed knots).
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 M5-branes in string theory (M-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)
Satoshi Nawata, Alexei Oblomkov, Lectures on knot homology (arXiv:1510.01795)