M Theory Lesson 25

In Bar-Natan’s talk about Khovanov homology he works out the Jones polynomial for a trefoil knot.

This means considering all smoothings of the 3-crossing knot, which are labelled by the ordered triplets 000, 100, 010, 001, 110, 101, 011 and 111. The 111 stands for the final result of three separate unlinked loops. The term 110 is a two loop diagram, contributing $q^{2} (q + q^{-1})^{2}$ to the invariant. The power weighting of the $q^{2}$ factor out the front goes with the number of disjoint loops. The 000 term is a bit different, because it corresponds to one loop inside another. It contributes $(q + q^{-1})^{2}$ to the invariant. There is a factor of $(q + q^{-1})$ for each loop.

Recall that Pfeiffer and Lauda studied this homology theory in the context of a 2-categorical TFT for open and closed strings.

2 Responses so far »

  1. 1

    Doug said,

    The first paper [karoubi.pdf] on the Dror Bar-Natan link has a catenoid in figure 2.2.

    Look at the ‘Double Bubble’, especially the right side.
    This is a torus [with paraboloid curvature?] around a catenoid with closed ends.
    http://mathworld.wolfram.com/DoubleBubble.html

    The second paper listed, Figure 1 Delooping: ‘Fast Khovanov Homology Computations’ has a general appearance similar to Petri Nets.

    This English version of the French ‘Max Plus Algebra Home Page’ is a gateway to many related papers, including Petri Nets, monoids and dioids, etc and VN KOLOKOLTSOV, VP MASLOV, ‘Idempotent Analysis and its Applications to Optimal Control ’ (in Russian) , Nauka 1994.
    http://www-rocq.inria.fr/MaxplusOrg/

  2. 2

    Kea said,

    Thanks, Doug! I haven’t looked through all the links myself – yes, it seems he has an interesting perspective on things!


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