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Algebraic Aspects of Heegaard Floer Theory

Reading Group@Université de Sherbrooke 2017

D3-2031, Tuesdays from 14:30 to 15:50

This term, we are reading Ozsváth and Szabó's recent paper entitled "Bordered knot algebras with matchings"[OS17]. In this paper, Ozsváth and Szabó give a refinement of a knot invariant defined in an earlier paper[OS16] using a divide-and-conquer strategy. First, they slice a knot diagram into elementary pieces with which they associate various bimodules, the types of which are familiar from bordered Heegaard Floer theory. Then, the knot invariant is defined as a certain tensor product\(~\boxtimes\) between those bimodules. Ozsváth and Szabó conjecture that this new knot invariant agrees with knot Floer homology. However, this construction is purely algebraic, so in particular, there is no mention of holomorphic curves. Since invariance is proved locally, a tangle invariant is implicit in their construction.

The goal of this reading group is to get a working understanding of the construction so as to be able to compute the invariants in a number of examples. In particular, 4-ended tangles such as some rational tangles, 2-stranded pretzel tangles, etc. would be interesting to compute, as there should be some link to the peculiar invariants from [Z16].

As a follow-up, it would also be interesting to understand the decategoried story. There is a fairly obvious interpretation (just take the Euler characteristic for each idempotent), but also a perhaps less-obvious one via the representation theory of \(\mathcal{U}_q(\mathfrak{gl}(1\vert 1))\), which has been studied by Manion[M16].

  • September 12 (Claudius)
    • Overview and context[OS02,R03,OS05]
    • Alexander polynomial via Kauffman states[K86,Z16]
    • The algebras \(\mathcal{B}(m,k)\) and \(\mathcal{B}_0(m,k)\)[OS17, section 2.1]
  • September 19 (Claudius)
    • Example computations for algebras \(\mathcal{B}(m,k)\) and \(\mathcal{B}_0(m,k)\)
    • Hey, I know this Fukaya cat! (\(\mathcal{B}(m,1)\))
  • September 26 (Claudius)
    • Algebraic structures from dg cats[Z16, appendix A]
  • October 3 (Claudius)
    • The algebras \(\mathcal{A}(m,k)\) and \(\mathcal{A'}(m,k)\)[OS17, section 2.2, 2.5, 2.6]
    • grading on algebras[OS17, section 2.7]
    • type DD bimodules for a crossing[OS17, section 3.1]
  • October 10
    • type DD bimodules for a maximum[OS17, section 5.1]
    • type DA bimodules for a maximum[OS17, section 5.2]
  • October 17
    • type DD bimodules for a crossing[OS17, section 3.2]
    • Computation for Reidemeister I: DA(maximum)\(\boxtimes\)DA(crossing)[OS17, section 5.2]
  • October 24
    • First calculations for rational tangles, e.g. for \(n\)-twists
    • Computation for Reidemeister I: DA(maximum)\(\boxtimes\)DA(crossing)[OS17, section 5.2]

References

[K83] L. Kauffman, Formal Knot Theory, Princeton University Press (1983)

[M16] A. Manion, On the decategorification of Ozsváth and Szabó's bordered theory for knot Floer homology

[OS02] P. Ozsváth and S. Szabó, Holomorphic disks and knot invariants, Adv. Math. 186 (2004), no. 1, 58-116

[OS05] —, Holomorphic disks and link invariants, Algebr. Geom. Topol. 8 (2008), 615-692

[OS16] —, Kauffman states, bordered algebras, and a bigraded knot invariant

[OS17] —, Bordered knot algebras with matchings

[R03] J. A. Rasmussen, Floer homology and knot complements, PhD thesis (2003), Harvard

[Z16] C. B. Zibrowius, On a Heegaard Floer homology for 4-ended tangles, PhD thesis (2017), Cambridge

[APT.m] —, APT.m, Mathematica package for computing Alexander polynomials of tangles, ancilliary file for [Z16].

last updated on the 19th September 2017
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