nLab knot invariant

Idea

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).

Examples

• crossing number
• bridge number
• unknotting number
• colorability
• knot group= the fundamental group of the complement of the knot.
• quandles and racks
• knot genus?
• Jones polynomial
• HOMFLY polynomial
• Alexander polynomial
• Reshetikhin-Turaev invariants for any object in a ribbon category
• Khovanov homology
• Kauffman bracket? (an invariant of framed links closely related to the Jones polynomial)

Many of these extend to link invariants or have variants that depend on the knot being oriented.

References

General

(…)

Knot invariants via topological strings and 5-branes

On realization of knot invariants/knot homology via topological string theory and BPS states:

• Edward Witten, Chern-Simons gauge theory as a string theory, in: The Floer memorial volume, Progr. Math. 133, Birkhäuser (1995) 637-678 [doi:10.1007/978-3-0348-9217-9, arXiv/hep-th/9207094, MR97j:57052]

• Hirosi Ooguri, Cumrun Vafa: Knot Invariants and Topological Strings, Nucl. Phys. B 577 (2000) 419-438 [doi:10.1016/S0550-3213(00)00118-8, arXiv:hep-th/9912123]

• Sergei Gukov, Albert Schwarz, Cumrun Vafa: Khovanov-Rozansky Homology and Topological Strings, Lett. Math. Phys. 74 (2005) 53-74 [doi:10.1007/s11005-005-0008-8arXiv:hep-th/0412243]

• Sergei Gukov: Surface Operators and Knot Homologies, Fortschritte der Physik 55 5-7 (2007) 473-490 [doi:10.1002/prop.200610385, arXiv:0706.2369]

• Mina Aganagic, Cumrun Vafa, Large $N$ duality, mirror symmetry, and a Q-deformed A-polynomial for knots [arXiv:1204.4709]

Understanding this via NS5-branes/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 16 3 (2012) [doi:10.4310/ATMP.2012.v16.n3.a5, arxiv:1106.4789]

• Edward Witten: Khovanov Homology And Gauge Theory, Geometry & Topology Monographs 18 (2012) 291-308 [pdf, arXiv:1108.3103]

• Sergei Gukov, Marko Stošić: Homological algebra of knots and BPS states, Geometry & Topology Monographs 18 (2012) 309-367 [doi:10.2140/gtm.2012.18.309, arXiv:1112.0030]

Review:

• Edward Witten, Khovanov Homology And Gauge Theory, Clay Conference, Oxford (October 2013) $[$pdf]

• Ross Elliot, Sergei Gukov: Section 1 of: Exceptional knot homology, Journal of Knot Theory and Its Ramifications 25 03 (2016) 1640003 [doi:10.1142/S0218216516400034, arXiv:1505.01635]

• Satoshi Nawata, Alexei Oblomkov: Lectures on knot homology, in: Physics and Mathematics of Link Homology, Contemp. Math. 680 (2016) 137 [doi:10.1090/conm/680, arXiv:1510.01795]

An alternative approach:

• Márk Mezei, Silviu S. Pufu, Yifan Wang: Chern-Simons theory from M5-branes and calibrated M2-branes, J. High Energ. Phys. 2019 165 (2019) [doi:10.1007/JHEP08(2019)165, arXiv:1812.07572]

Via D$p$/D$(p+2)$ bound states / monopoles

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]

category: knot theory