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knot invariant

Contents

Contents

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

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

References

General

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In string theory

Knot invariants arising in string theory/M-theory:

Via 5-brane BPS states

Discussion of knot invariants in terms of BPS states of M5-branes:

Via Dpp/D(p+2)(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/N1/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 NN 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

Last revised on November 6, 2019 at 14:33:31. See the history of this page for a list of all contributions to it.