Definitions, diagrams, Reidemeister moves, wild knots, connect sum, unknotting, satellite knots, historical overview of knot theory.

Braid group, Alexander and Markov theorems, braid index. Rational tangles, rational links, 2-bridge links, plats.

Seifert surfaces, circuits and genus, Yamada's theorem and Vogel's algorithm. Seifert matrix, S-equivalence. Signature and determinant.

Many approaches to Alexander polynomial: Seifert matrices, homology of infinite cyclic cover, Fox calculus, Conway polynomial. Wirtinger presentation for the knot group.

Kauffman bracket, state sums. Alternating knots and proof of Tait's conjecture. Adequate diagrams. Turaev surface and genus.

Tait graph and spanning-tree expansion for the Jones polynomial. Tutte polynomial of graphs. Kauffman bracket skein module. Kanenobu links.

Representations, R-matrices and Yang-Baxter equation, colored Jones polynomial, cabling formula.

Calculations with Jones-Wenzl idempotents, colored Jones polynomial.

Boundary slopes, Slope Conjecture(s), proof for adequate knots.

Hyperbolic geometry of 3-manifolds, figure-8 knot complement, ideal tetrahedra, gluing and completeness equations.

Hyperbolic volume, properties, diagrammatic bounds.

Proof for figure-8 knot, support for the Volume Conjecture.

Approaches to Khovanov homology: Khovanov, Viro, Bar-Natan, spanning tree model, Turaev genus and homological width.

Lickorish:

Murasugi:

Prasolov-Sossinsky:

Purcell:

Rolfsen:

Good introductory texts:
*The knot book* by Colin Adams

*Knots Knotes* by Justin Roberts

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.

An introduction to knot Floer homology, by Ciprian Manolescu.

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

SnapPy.

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.