An introduction to knot theory
Seminar topics for the examination

Exams will be held on April 10, 2019, starting from 10, in Salle au Val. Here is the page of the course.

This is a list of possible topics, with references:

• Intrinsic knotting: every embedding of the complete graph on 6 (resp. 7) vertices contains a non-trivial link (resp. knot) [CG].


• Branched double covers and the Goeritz matrix: or, how non-orientable surfaces still present the homology of the double cover, and the resulting skein relation on the determinant of a knot/link [Lic, Chapter 9].


• The HOMFLY-PT polynomial: the HOMFLY-PT polynomial is well-defined and its relation to the canonical genus [Lic, Theorems 15.2 and 16.7].


• Pass-moves and the Arf invariant: every knot is pass-equivalent (this is a certain diagramatic operation) either to the unknot or the trefoil, but not both [Kau, Chapter 5].


• The Alexander polynomial and alternating links: alternating links have alternating Alexander polynomial, and moreover their genus is detected by it [Cro].


• Knot polynomials of periodic knots: periodic knots are knots with a cyclic symmetry; their polynomials, too, have some structure [Prz1].


• Quadrisecants and the Fary–Milnor theorem: a quadrisecant of a knot is a straight line meeting the knot in four points; all non-trivial knots have a(n alternating) quadrisecant, and this can be used to prove that the total curvature of a knot is always larger than 4π [MM].


• Invariance of Khovanov homology: the name says it all [Tur, Lectures 1–2].


• Bureau representation and the Alexander polynomial: from braid representations to the Alexander polynomial (the topological way) [Mor1].


• 3-colourings and the Jones polynomial: 3-colourings are encoded in the Jones polynomial [Prz2].


• Alexander's and Markov's theorems: the proofs of braid to knots, and back [Mor2].


• The Poincaré sphere: the Poincaré is one of the first "interesting counterexamples" in topology; the goal of the talk would be to give several different presentations for it, including a concrete calculation of its fundamental group [PS, Section 18].

[CG] J. H. Conway and C. McA. Gordon, Knots and links in spatial graphs, doi:10.1002/jgt.3190070410.
[Cro] R. H. Crowell, Genus of alternating knot types, doi:10.2307/1970181.
[Kau] L. Kauffman, On knots, Princeton University Press.
[Lic] R. W. B. Lickorish, An introduction to knot theory, Springer.
[Mor1] H. R. Morton, The multivariable Alexander polynomial for a closed braid, doi:10.1090/conm/233/03427.
[Mor2] H. R. Morton, Threading knot diagrams, doi:10.1017/S0305004100064161.
[MM] H. R. Morton and D. M. Q. Mond, Closed curves with no quadrisecants, doi:10.1016/0040-9383(82)90007-6.
[PS] V. V. Prasolov and A. B. Sossinsky, Knot, links, braids, and 3-manifolds, American Mathematical Society.
[Prz1] J. H. Przytycki, On Murasugi's and Traczyk's criteria for periodic links, eudml.org/doc/164519.
[Prz2] J. H. Przytycki, 3-Coloring and other Invariants of Knots, in Knot theory, Banach center publications.
[Tur] P. Turner, Five lectures on Khovanov homology, doi:10.1142/S0218216517410097.

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