William D. Gillam

COMPUTATIONS OF HEEGAARD-FLOER KNOT HOMOLOGY

We compute the knot Floer homology of knots with at most 12 crossings, as well as the τ invariant for knots with at most 11 crossings, using the combinatorial approach described by Manolescu, Ozsvath and Sarkar. We review their construction, giving two examples that can be workout out by hand, and we explain some ideas we used to simplify the computation. We conclude with a discussion of knot Floer homology for small knots, and we formulate a conjecture about the behavior of knot Floer homology under mutation, paying especially close attention to the Kinoshita–Terasaka knot and its Conway mutant. Finally, we discuss a conjecture of Rasmussen on relationship between Khovanov homology and knot Floer homology, and observe that it is consistent with our calculations.

Cusp size bounds from singular surfaces in hyperbolic 3-manifolds

Singular maps of surfaces into a hyperbolic 3-manifold are utilized to find upper bounds on meridian length, l-curve length and maximal cusp volume for the manifold. This allows a proof of the fact that there exist hyperbolic knots with arbitrarily small cusp density and that every closed orientable 3-manifold contains a knot whose complement is hyperbolic with maximal cusp volume less than or equal to 9. We also find particular upper bounds on meridian length, l-curve length and maximal cusp volume for hyperbolic knots in S 3 depending on crossing number. Particular improved bounds are obtained for alternating knots.

KNOT HOMOLOGY OF (3, m) TORUS KNOTS

We give a direct computation of the Khovanov knot homology of the (3, m) torus knots/links. Our computation yields complete results with ℤ[½] coefficients, though we leave a slight ambiguity concerning 2-torsion when integer coefficients are used. Our computation uses only the basic long exact sequence in knot homology and Rasmussens result on the triviality of the embedded surface invariant.

A sheaf-theoretic description of Khovanov's knot homology

We give a description of Khovanovs knot homology theory in the language of sheaves. To do this, we identify two cohomology theories associated to a commutative diagram of abelian groups indexed by elements of the cube {0, 1}n. The first is obtained by taking the cohomology groups of the chain complex constructed by summing along the diagonals of the cube and inserting signs to force d2 = 0. The second is obtained by regarding the commutative diagram as a sheaf on the cube (in the order-filter topology) and considering sheaf cohomology with supports. Included is a general study of sheaves on finite posets, and a review of some basic properties of knot homology in the language of sheaves.