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The traditional skein relation for the Alexander polynomial involves an oriented knot, K+, with a distinguished positive crossing; a knot K−, obtained by changing the distinguished positive crossing of K+ to a negative crossing; and a link K0, the orientation preserving resolution of the distinguished crossing. We refer to (K+,K−,K0) as the oriented skein triple.
A tangle is defined as a pair (B, t) of a 3-dimensional ball B and a collection of disjoint, simple, properly embedded arcs, denoted t. DeWitt Sumners and Claus Ernst developed the tangle model which uses the mathematics of tangles to model DNA-protein binding. The protein is seen as the 3-ball and the DNA bound by the protein as properly embedded curves in the 3-ball. Topoisomerases are proteins that break one segment of DNA allowing a DNA segment to pass through before resealing the break. Effectively, the action of these proteins can be modeled as K− ↔ K+. Recombinases are proteins that cut two segments of DNA and recombine them in some manner. While recombinase local action varies, most are mathematically equivalent to a resolution, i.e. K± ↔ K0. The oriented triple is now viewed as K− = circular DNA substrate, K+ = product of topoisomerase action, K0 = product of recombinase action.
The theorem stated in this dissertation gives a relationship between two 2-bridge knots, K+ and K−, that differ by a crossing change and a link, K0 created from the oriented resolution of that crossing. We apply this theorem to difference topology experiments using topoisomerase proteins to study SMC proteins.
In recent years, link homology theories have become a popular invariant to develop and study. One such invariant knot Floer homology, was constructed by Peter Ozsváth, Zoltán Szabó, and independently Jacob Rasmussen, denoted by HFK. It is also a refinement of a classical invariant, the Alexander polynomial.
The study of DNA knots and links are of great interest to molecular biologists as they are present in many cellular process. The variety of experimentally observed DNA knots and links makes separating and categorizing these molecules a critical issue. Thus, knowing the knot Floer homology will provide restrictions on knotted and linked products of protein action.
We give a summary of the combinatorial version of knot Floer homology from known work, providing a worked out example. The thesis ends with reviewing knot Floer homology properties of three particular sub-families of biologically relevant links known as (2, p)- torus links, clasp knots and 3-strand pretzel links.