Department of Mathematcs, The Graduate Center, City University of New York (CUNY)

MATH 82800: Knot Theory [23783]

Prof. Ilya Kofman

GC Office:   4214.08
Email:   ikofmanmath.csi.cuny.edu
Website:   http://www.math.csi.cuny.edu/~ikofman/

Course outline

  • 1. Knots and links
    Definitions, diagrams, Reidemeister moves, wild knots, connect sum, unknotting, linking number, Dowker-Thistlethwaite code, satellite knots, famous families of knots, historical overview of knot theory.
    Reading:  Lickorish ch 1, Cromwell ch 1-4, Prasolov-Sossinsky ch 1.

  • 2. Braids and tangles
    Braid group, Alexander and Markov theorems, braid index. Rational tangles, rational links, 2-bridge links, plats.
    Reading:  Prasolov-Sossinsky ch 3, Cromwell ch 8, 10.1, 10.4.

  • 3. Seifert surfaces and Seifert matrices
    Seifert surfaces, circuits and genus, Yamada's theorem and Vogel's algorithm. Seifert matrix, S-equivalence. Signature and determinant.
    Reading:  Lickorish ch 2,8, Cromwell ch 5-6, Murasugi ch 5, 6.4.

  • 4. Alexander polynomial
    Many approaches to Alexander polynomial: Seifert matrices, homology of infinite cyclic cover, Fox calculus, Conway polynomial. Wirtinger presentation for the knot group.
    Reading:  Lickorish ch 6,7,11, Cromwell ch 7, Rolfsen ch 6-8, Murasugi ch 6.

  • 5. Jones polynomial I
    Kauffman bracket, state sums. Alternating knots and proof of Tait's conjecture. Adequate diagrams. Turaev surface and genus.
    Reading:  Lickorish ch 3,5, Prasolov-Sossinsky ch 2.3, Cromwell ch 9.

  • 6. Jones polynomial II
    Tait graph and spanning-tree expansion for the Jones polynomial. Tutte polynomial of graphs. Temperley-Lieb algebra and braid representations.
    Reading:  Bollobas Modern Graph Theory ch X, Jones 1, Jones 2.

  • 7. Linear skein theory
    HOMFLYPT polynomial. Kauffman bracket skein module. Skein linear algebra to get links with trivial Jones polynomial. Kanenobu links. How twisting strands affects the Jones polynomial.
    Reading:  Prasolov-Sossinsky ch 8, Watson, Rolfsen.

  • 8. Jones-Wenzl idempotents
    Calculations with Jones-Wenzl idempotents, colored Jones polynomial.
    Reading:  Lickorish ch 13-14, Prasolov-Sossinsky ch 8, Masbaum-Vogel

  • 9. Quantum invariants
    Representations, R-matrices and Yang-Baxter equation, colored Jones polynomial, cabling formula.
    Reading:  Chmutov, Duzhin, Mostovoy Vassiliev Knot Invariants ch 2.6 and appendix. Ohtsuki Quantum Invariants ch 4. H.Murakami, "An introduction to the Volume Conjecture".

  • 10. Hyperbolic knots
    Hyperbolic geometry of 3-manifolds, figure-8 knot complement, ideal tetrahedra, gluing and completeness equations.
    Reading:  Purcell, Bonahon Low-dimensional geometry, Adams, Weeks.

  • 11. Hyperbolic volume
    Hyperbolic volume, properties, diagrammatic bounds. SnapPy.
    Reading:  Ratcliffe Foundations of hyperbolic manifolds chapter 10, Milnor, Lackenby-Agol-DThurston. SnapPy.

  • 12. Volume Conjecture
    Proof for figure-8 knot, support for the Volume Conjecture.
    Reading:  H.Murakami "An introduction to the Volume Conjecture".

  • 13. Khovanov homology
    Approaches to Khovanov homology: Khovanov, Viro, Bar-Natan, spanning tree model, Turaev genus and homological width.
    Reading:  P.Turner, "Five lectures on Khovanov homology."

  • 14. Knot Floer homology
    Special class by Robert Lipshitz (Columbia), Tuesday May 20, 11am-1pm, Room 3212.
    Reading:  C.Manolescu, "An introduction to knot Floer homology."

    Textbooks

    Cromwell:  Knots and links by Peter Cromwell
    Lickorish:  An introduction to knot theory by W.B.Raymond Lickorish.
    Murasugi:  Knot theory by Kunio Murasugi
    Prasolov-Sossinsky:  Knots, links, braids and 3-manifolds by V.V.Prasolov and A.B.Sossinsky
    Rolfsen:  Knots and links by Dale Rolfsen

    Good introductory texts:
    The knot book by Colin Adams
    Knots Knotes by Justin Roberts

    Other resources

    Handbook of Knot Theory (W.Menasco and M.Thistlethwaite, editors)
    1. Colin Adams, Hyperbolic knots
    2. Joan S. Birman and Tara Brendle, Braids and knots
    3. John Etnyre, Legendrian and transversal knots
    4. Cameron Gordon, Dehn surgery
    5. Jim Hoste, The enumeration and classiffication of knots and links
    6. Louis Kauffman, Diagrammatic methods for invariants of knots and links
    7. Charles Livingston, A survey of classical knot concordance
    8. Marty Scharlemann, Thin position
    9. Lee Rudolph, Knot theory of complex plane curves
    10. DeWit Sumners, The topology of DNA
    11. Jeff Weeks, Computation of hyperbolic structures in knot theory

    Two Lectures On The Jones Polynomial And Khovanov Homology, by Edward Witten. An advanced gauge theory approach.

    The Trieste look at knot theory, by Jozef Przytycki. Introduction to knots and a survey of knot colorings.
    3-coloring and other elementary invariants of knots, by Jozef Przytycki.

    Short review of braids, by Dale Rolfsen.
    An Introduction to braid theory, by Maurice Chiodo.
    An elementary introduction to the theory of braids, by Roger Fenn.
    About the Temperley-Lieb algebra: by V.S.Sunder, Anne Moore, Dana Emst.

    Topological invariants of knots: three routes to the Alexander polynomial, by Edward Long.
    Knot theory and the Alexander polynomial, by Reagin McNeill.

    Data on knots and their invariants:
    The Knot Atlas (wiki), by Dror Bar-Natan and Scott Morrison. Among other info, it contains Rolfsen's table of knots up to 10 crossings.
    Table of Knot Invariants, by Charles Livingston and Jae Choon Cha.
    The KnotPlot Site

    The CompuTop.org Software Archive, a site for people doing computational stuff with low-dimensional topology.

    Other knot theory books:

    Knot theory, a book by Vassily Manturov.
    On knots, by Louis Kauffman;
    Formal knot theory, by Louis Kauffman;
    Knots and physics, by Louis Kauffman;
    Knot theory, by Charles Livingston;
    Introduction to knot theory, by R.Crowell and R.Fox;
    A survey of knot theory, by A.Kawauchi;
    Braids, links and mapping class groups, by Joan Birman.