GLOBAL MUTATION OF KNOTS

Many polynomial invariants of knots and links, including the Jones and HOMFLY-PT polynomials, are widely used in practice but #P-hard to compute. It was shown by Makowsky in 2001 that computing the Jones polynomial is fixed-parameter tractable in the treewidth of the link diagram, but the parameterised complexity of the more powerful HOMFLY-PT polynomial remained an open problem. Here we show that computing HOMFLY-PT is fixed-parameter tractable in the treewidth, and we give the first sub-exponential time algorithm to compute it for arbitrary links.

We introduce an infinite family of quantum enhancements of the biquandle counting invariant which we call biquandle virtual brackets. Defined in terms of skein invariants of biquandle colored oriented knot and link diagrams with values in a commutative ring [Formula: see text] using virtual crossings as smoothings, these invariants take the form of multisets of elements of [Formula: see text] and can be written in a “polynomial” form for convenience. The family of invariants defined herein includes as special cases all quandle and biquandle 2-cocycle invariants, all classical skein invariants (Alexander–Conway, Jones, HOMFLYPT and Kauffman polynomials) and all biquandle bracket invariants defined in [S. Nelson, M. E. Orrison and V. Rivera, Quantum enhancements and biquandle brackets, J. Knot Theory Ramifications 26(5) (2017) 1750034] as well as new invariants defined using virtual crossings in a fundamental way, without an obvious purely classical definition.

The extreme degrees of the colored Jones polynomial of any link are bounded in terms of concrete data from any link diagram. It is known that these bounds are sharp for semi-adequate diagrams. One of the goals of this paper is to show the converse; if the bounds are sharp then the diagram is semi-adequate. As a result, we use colored Jones link polynomials to extract an invariant that detects semi-adequate links and discuss some applications.

Khovanov homology and knot Floer homology are generalizations of the Jones polynomial and the Alexander polynomial respectively. They are bigraded Z-modules, and their underlying polynomials are recovered by taking the graded Euler characteristic. The two homologies share many characteristics, however their relationship has yet to be fully understood. In both Khovanov homology and knot Floer homology, the two gradings can be combined into a single diagonal grading. Homological width is a measure of the support of the homology with respect to the diagonal grading. In this thesis, we show that the homological width of Khovanov homology and knot Floer homology have a common upper bound. Every link diagram has an associated Turaev surface, a certain Heegaard surface in the 3-sphere on which the knot has an alternating projection. The Turaev genus of a knot is the minimum genus of a Turaev surface where the minimum is taken over all diagrams of the knot. Turaev introduced this surface in order to prove a conjecture about the span of the Jones polynomial. Previously, it has been shown that Turaev genus gives an upper bound for the homological width of Khovanov homology. Since Khovanov homology is a generalization of the Jones polynomial, one might expect that Turaev genus and Khovanov homology are related. In this thesis, we show that Turaev genus also gives an upper bound for the homological width of knot Floer homology, giving the first known relationship between the Alexander polynomial and the Turaev surface. It is also more evidence towards a relationship between Khovanov homology and knot Floer homology. In addition, we construct infinite families of links whose Khovanov homology have the same homological width. Using this construction, we compute the Khovanov width of all closed 3-braids.

This paper is an introduction to disoriented knot theory, which is a generalization of the oriented knot and link diagrams and an exposition of new ideas and constructions, including the basic definitions and concepts such as disoriented knot, disoriented crossing and Reidemesiter moves for disoriented diagrams, numerical invariants such as the linking number and the complete writhe, the polynomial invariants such as the bracket polynomial, the Jones polynomial for the disoriented knots and links.

The second author previously discussed how classical complexity separation conjectures, we call them “axioms,” have implications in three manifold topology: polynomial length strings of operations which preserve certain Jones polynomial evaluations cannot produce exponential simplications of link diagrams. In this paper, we continue this theme, exploring now more subtle separation axioms for quantum complexity classes. Surprisingly, we now find that similar strings of operations are unable to effect even linear simplications of the diagrams.

Every Kauffman state \sigma of a link diagram D(K) naturally defines a state surface S_\sigma whose boundary is K. For a homogeneous state \sigma, we show that K is a fibered link with fiber surface S_\sigma if and only if an associated graph G'_\sigma is a tree. As a corollary, it follows that for an adequate knot or link, the second and next-to-last coefficients of the Jones polynomial are obstructions to certain state surfaces being fibers for K. This provides a dramatically simpler proof of a theorem from [arXiv:1108.3370].