**Summer 2019 Tutorial: Knot Invariants**

**Meetings:** MWTh, 6:30 - 8:30pm, SC 232. (Lectures M/W and problem session Th.)

** Instructors:** Morgan Opie and Joshua Wang

**Office:** SC 505g (Morgan) and 425a (Josh)

**Emails:** opie AT g DOT harvard DOT edu

jxwang AT math DOT harvard DOT edu

**Update: since July 4 is a holiday, there will be no problem session that evening. Instead, Morgan will hold office hours/ problem session in the 505 alcove from 3-5 pm on July 5, 2019.**

**Course summary.**
We’ll begin the tutorial with an introduction to knot theory. We’ll discuss a few classical knot invariants (genus, unknotting number, slice genus) with many pictures and examples, culminating with an elementary diagramatic treatment of the Jones polynomial.

The rest of the tutorial will focus on two generalizations of the Jones polynomial, both having a categorical/algebraic flavor. The first will be Khovanov homology, a theory which “categorifies” the Jones polynomial in the same way that singular homology categorifies Euler characteristic. The Euler characteristic of a (nice enough) space is an integer. It is a fairly useful invariant, but it is not functorial: Euler characteristic does not associate anything to a continuous map between spaces. Singular homology is its “categorification”; it associates to a space a sequence of abelian groups, and to
every continuous map a sequence of group homomorphisms. Euler characteristic is recovered
by taking an alternating sum of the ranks of these abelian groups. Khovanov homology is to
the Jones polynomial just as singular homology is to Euler characteristic. We will see that the Jones polynomial can be recovered from Khovanov homology by taking a suitable alternating sum of ranks, and that there is a category whose objects are knots with respect to which Khovanov homology is functorial.

The second generalization is to the knot polynomials defined by Reshetikhin and Turaev using ribbon categories. It is a current area of research to categorify these knot
polynomials just as Khovanov homology has done for the Jones polynomial. Our main goal will be to recover the Jones polynomial in this framework, and we will develop the necessary machinery to obtain this goal along the way.

We do not expect students to have any prior exposure to knot theory or category theory, but having familiarity with basic algebraic topology will be very helpful.

For general information about Harvard tutorials, see here.

** Timeline. **

The tutorial will run ** June 10 - July 29, with a break July 7-15.**

**June 10-27**: Josh delivers the first three weeks of lectures. Topics include on basic concepts of knot theory, ways to formalize the intuitive idea of a knot, Reidermeister moves, knot invariants (including unknotting number, tricolorability, genus), the Jones polynomial, and Khovanov homology. Among other things, also discuss classification of (abstract) surfaces, surfaces in 3- and 4- space (and how these are related to the study of knot invariants), four-ball genus and its relationship to unknotting number, and the Milnor conjecture and some concepts behind its proof.

**July 1-7, 17-29**: Morgan takes over as lecturer. We’ll leave behind Khovanov homology with the goal of finding another generalization of the Jones polynomial. We’ll begin with a week on relevant category theory (basic category theory, monodical categories, ribbon categories). This will include two lectures (July 1 and 3) and one problem session. After a summer break (July 7-15) we will resume with lectures and problem sessions July 17-29. Our focus for these two weeks will be to understand the generalization of the Jones polynomial given in this paper of Reshetikhin and Turaev.

**Course materials.**

Worksheet 1

Worksheet 2

Worksheet 3

Worksheet 4

Worksheet 5

Worksheet 6

Worksheet 7

Worksheet 8

Worksheet 9

Worksheet 10

Worksheet 11

Copy of Rolfsen’s table of knots

Background by Brirush - Own work, CC BY-SA 3.0, Link