nLab link invariant

Idea

Many of the knot invariants have analogues for links. Some, in fact seem to be better thought of as being restrictions of link invariants to single component links (i.e. knots). We repeat the statement from knot invariant with obvious adjustments.

Definition

A link invariant is map from isotopy classes of links to any kind of structure you could imagine. These are helpful because it’s much easier to check that the structures one maps to (numbers, groups, etc.) are different than it is to check that links are different.

Examples

• Linking number

References

General

See the references at knot invariant.

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]