Proof for figure-8 knot, support for the Volume Conjecture. Reading: Purcell ch 9, Ratcliffe Foundations of hyperbolic manifolds chapter 10, Milnor, Lackenby-Agol-DThurston. Hyperbolic volume, properties, diagrammatic bounds. Hyperbolic geometry of 3-manifolds, figure-8 knot complement, ideal tetrahedra, gluing and completeness equations. Reading: slope conj (Garoufalidis), adequate knots (FKP) and survey (FKP), Montesinos boundary slopes (Dunfield), boundary slopes (Culler-Shalen), slope conj for links (R.v.Veen), strong slope conj (Kalfagianni-Tran) Reading: Lickorish ch 13-14, Prasolov-Sossinsky ch 8, Masbaum-Vogelīoundary slopes, Slope Conjecture(s), proof for adequate knots. H.Murakami, "An introduction to the Volume Conjecture".Ĭalculations with Jones-Wenzl idempotents, colored Jones polynomial. Reading: Chmutov, Duzhin, Mostovoy Vassiliev Knot Invariants ch 2.6 and appendix. Representations, R-matrices and Yang-Baxter equation, colored Jones polynomial, cabling formula. Reading: Bollobas Modern Graph Theory ch X, Jones 1, Jones 2, Watson, Rolfsen. Tait graph and spanning-tree expansion for the Jones polynomial. Reading: Lickorish ch 3,5, Prasolov-Sossinsky ch 2.3, Cromwell ch 9. Alternating knots and proof of Tait's conjecture. Reading: Lickorish ch 6,7,11, Cromwell ch 7, Rolfsen ch 6-8, Murasugi ch 6. Wirtinger presentation for the knot group. Many approaches to Alexander polynomial: Seifert matrices, homology of infinite cyclic cover, Fox calculus, Conway polynomial. Reading: Lickorish ch 2,8, Cromwell ch 5-6, Murasugi ch 5, 6.4. Seifert surfaces, circuits and genus, Yamada's theorem and Vogel's algorithm. Reading: Prasolov-Sossinsky ch 3, Cromwell ch 8, 10.1, 10.4, Rolfsen braids. Rational tangles, rational links, 2-bridge links, plats. Reading: Lickorish ch 1, Cromwell ch 1-4, Prasolov-Sossinsky ch 1, Knotes ch 1-2, Lackenby survey.īraid group, Alexander and Markov theorems, braid index.
Now that we have Reidemeister's theorem, we can at last construct some invariants and use them to prove that certain knots and links are inequivalent.The Graduate Center, City University of New York (CUNY) MATH 82800: Knot Theory ĭefinitions, diagrams, Reidemeister moves, wild knots, connect sum, unknotting, satellite knots, historical overview of knot theory. A knot invariant is any function i of knots which depends only on their equivalence classes. We have not yet speci ed what kind of values an invariant should take. The most common invariants are integer-valued, but they might havevalues in the rationals Q, a polynomial ring Z, a Laurent polynomial ring (negative powers of x allowed) Z, or even be functions which assign to any knot a group (thought of up to isomorphism). The function of an invariant istodistinguish (i.e. The de nition says that if K = K 0 then i(K) =i(K 0 ). Therefore if i(K) 6= i(K 0 ) then K K 0 cannot be equivalent they have been distinguished by i. Warning: the de nition does not work in reverse: if two knots have equal invariants then they are not necessarily equivalent.
As a trivial example, the function i which takes the value 0 on all knots is a valid invariant but which is totally useless! Better examples will be given below. Link invariants, oriented link invariants, and so on (for all the di erent types of knotty things we might consider) are de ned and used similarly. The crossing number c(K) is the minimal number of crossings occurring in any diagram of the knot K. This is an invariant by de nition, but at this stage the only crossing number we can actually compute is that of the unknot, namely zero! Example 3.1.7. The number of components (L) of a link L is an invariant (since wiggling via -moves does not change it, it does depend only on the equivalence class of link). De ne the stick number of a knot to be the minimal number of arc segments with which it can be built. Show that the only knots with 4 or 5 arcs are unknots, and show thus that the trefoil has stick number 6. De ne the human number (!) of a knot to be the minimal number of people it takes (holding hands in a chain) to make the knot - what is it for the trefoil and gure-eight? 3.2. One of the simplest invariants that can actually be computed easily is the linking number of an oriented link. It is computed by using a diagram of the link, so we then have to use Reidemeister's theorem to prove that it is independent of this choice of diagram, and consequently really does depend only on the original link. Then the total linking number Lk(D) is obtained by taking half the sum, over all crossings, of contributions from each given by +1 ,1 if the two arcs involved in the crossing belong to di erent components of the link, and 0 if they belong to the same one.
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