Examples/classes:
Types
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.
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Knot invariants arising in string theory/M-theory:
Discussion of knot invariants in terms of BPS states of M5-branes:
Edward Witten, Fivebranes and Knots, Quantum Topology, Volume 3, Issue 1, 2012, pp. 1-137 (arXiv:1101.3216, doi:10.4171/QT/26)
Davide Gaiotto, Edward Witten, Knot Invariants from Four-Dimensional Gauge Theory, Advances in Theoretical and Mathematical Physics, Volume 16 (2012) Number 3 (arxiv:1106.4789)
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)
Relation of Dp-D(p+2)-brane bound states/Yang-Mills monopoles to knot invariants via chord diagrams:
S. Ramgoolam, B. Spence, S. Thomas, Section 3.2 of: Resolving brane collapse with $1/N$ corrections in non-Abelian DBI, Nucl. Phys. B703 (2004) 236-276 (arxiv:hep-th/0405256)
S. McNamara, Constantinos Papageorgakis, S. Ramgoolam, B. Spence, Appendix A of: Finite $N$ effects on the collapse of fuzzy spheres, JHEP 0605:060, 2006 (arxiv:hep-th/0512145)
Constantinos Papageorgakis, p. 161-162 of: On matrix D-brane dynamics and fuzzy spheres, 2006 (pdf)
Last revised on February 6, 2020 at 02:09:45. See the history of this page for a list of all contributions to it.