arXiv subjects

Lawrence Roberts

Publications and source records attributed to Lawrence Roberts.

7 recordsLinked to original sources

arXiv2014

The decategorification of bordered Khovanov homology

In two previous papers, the author showed how to decompose the Khovanov homology of a link $\mathcal{L}$ into the algebraic pairing of a type D structure and a type A structure (as defined in bordered Floer homology), whenever a diagram for $\mathcal{L}$ is decomposed into the union of two tangles. Since Khovanov homology is the categorification of a version of the Jones polynomial, it is natural to ask what the type A and type D structures categorify, and how their pairing is encoded in the decategorifications. In this paper, the author constructs the decategorifications of these two structures, in a manner similar to Ina Petkova's decategorification of bordered Floer homology and shows how they recover the Jones polynomial. We also give a new proof of the mutation invariance of the Jones polynomial which uses these decomposition techniques.

math.GT↗

arXiv2014

Planar algebras and the decategorification of bordered Khovanov homology

We give a simple, combinatorial construction of a unital, spherical, non-degenerate $\ast$-planar algebra over the ring $\mathbb{Z}[q^{1/2},q^{-1/2}]$. This planar algebra is similar in spirit to the Temperley-Lieb planar algebra, but computations show that they are different. The construction comes from the combinatorics of the decategorifications of the type A and type D structures in the author's previous work on bordered Khovanov homology. In particular, the construction illustrates how gluing of tangles occurs in his bordered Khovanov homology and its difference from that in Khovanov's tangle homology, without being encumbered by any extra homological algebra. It also provides a simple framework for showing that these theories are not related through a simple process, thereby confirming recent work of A. Manion. Furthermore, using Khovanov's conventions and a state sum approach to the Jones polynomial, we obtain new invariant for tangles in $\Sigma \times [-1,1]$ where $\Sigma$ is a compact, planar surface with boundary, and the tangle intersects each boundary cylinder in an even number of points. This construction naturally generalizes Khovanov's approach to the Jones polynomial.

math.GT↗

arXiv2013

On non-isotopic spanning surfaces for a class of arborescent knots

We use the methods of Hedden, Juhasz, and Sarkar to exhibit a set of arborescent knots that bound large numbers of non-isotopic minimal genus spanning surfaces. In particular, we describe a sequence of prime knots K_{n} which will bound at least 2^{2n-1} non-isotopic minimal spanning surfaces of genus n.

math.GT↗

arXiv2011

Totally Twisted Khovanov Homology

We define a variation of Khovanov homology with an explicit description in terms of the spanning trees of a link projection. We prove that this new theory is a link invariant and describe some of its properties. Finally, we provide some the results of some computer computations of the invariant.

math.GT↗

arXiv2008

Notes on the Heegaard-Floer Link Surgery Spectral Sequence

This largely technical paper is divided into two parts: part I: An account of P. Ozsvath and Z. Szabo's construction of the link surgery spectral sequence. There are no new results here, but this part slightly modifies and expands their proof and is included as an aid to part II. part II: Some modifications of the spectral sequence suitable for knot Floer homology; an exposition of the invariance of the spectral sequence, under a suitable equivalence, from alterations of the underlying bouquet; and an account of the morphisms induced on the spectral sequences by suitable four dimensional cobordisms and how they depend upon the four manifold. This is the author's effort to investigate a question at the end of the introduction to P. Ozsvath's and Z. Szabo's paper "On the Heegaard-Floer homology of double branched covers."

math.GT↗

arXiv2006

Rational blow-downs in Heegaard-Floer homology

The paper describes how known results in Heegaard-Floer homology apply to all known examples of rational blow-downs, and provides several new four dimensional pieces which could be exchanged while preserving some of the Ozsváth-Szabó four manifold invariants, when one of these pieces is found embedded in a smooth four manifold.

math.GT↗

arXiv2006

Heegaard-Floer homology and string links

We extend knot Floer homology to string links in D^{2} \times I and to d-based links in arbitrary three manifolds, without any hypothesis on the null-homology of the components. As for knot Floer homology we obtain a description of the Euler characteristic of the resulting homology groups (in D^{2} \times I) in terms of the torsion of the string link. Additionally, a state summation approach is described using the equivalent of Kauffman states. Furthermore, we examine the situtation for braids, prove that for alternating string links the Euler characteristic determines the homology, and develop similar composition formulas and long exact sequences as in knot Floer homology.

math.GT↗