Trivalent Graph isomorphism in polynomial time

It's important to design polynomial time algorithms to test if two graphs are isomorphic at least for some special classes of graphs. An approach to this was presented by Eugene M. Luks(1981) in the work \textit{Isomorphism of Graphs of Bounded Valence Can Be Tested in Polynomial Time}. Unfortunately, it was a theoretical algorithm and was very difficult to put into practice. On the other hand, there is no known implementation of the algorithm, although Galil, Hoffman and Luks(1983) shows an improvement of this algorithm running in $O(n^3 \log n)$. The two main goals of this master thesis are to explain more carefully the algorithm of Luks(1981), including a detailed study of the complexity and, then to provide an efficient implementation in SAGE system. It is divided into four chapters plus an appendix.Comment: 48 pages. It is a Master Thesi

A note on Khovanov-Rozansky sl2sl_2-homology and ordinary Khovanov homology

In this note we present an explicit isomorphism between Khovanov-Rozansky $sl_2$-homology and ordinary Khovanov homology. This result was originally stated in Khovanov and Rozansky's paper \cite{KRI}, though the details have yet to appear in the literature. The main missing detail is providing a coherent choice of signs when identifying variables in the $sl_2$-homology. Along with the behavior of the signs and local orientations in the $sl_2$-homology, both theories behave differently when we try to extend their definitions to virtual links, which seemed to suggest that the $sl_2$-homology may instead correspond to a different variant of Khovanov homology. In this paper we describe both theories and prove that they are in fact isomorphic by showing that a coherent choice of signs can be made. In doing so we emphasize the interpretation of the $sl_2$-complex as a cube of resolutions.Comment: 19 pages, 11 figures. Expanded introduction and abstract. Remark added to end of section 4.

The loop expansion of the Kontsevich integral, the null move and S-equivalence

This is a substantially revised version. The Kontsevich integral of a knot is a graph-valued invariant which (when graded by the Vassiliev degree of graphs) is characterized by a universal property; namely it is a universal Vassiliev invariant of knots. We introduce a second grading of the Kontsevich integral, the Euler degree, and a geometric null-move on the set of knots. We explain the relation of the null-move to S-equivalence, and the relation to the Euler grading of the Kontsevich integral. The null move leads in a natural way to the introduction of trivalent graphs with beads, and to a conjecture on a rational version of the Kontsevich integral, formulated by the second author and proven in joint work of the first author and A. Kricker.Comment: AMS-LaTeX, 20 pages with 31 figure

Categorifying Hecke algebras at prime roots of unity, part I

We equip the type $A$ diagrammatic Hecke category with a special derivation, so that after specialization to characteristic $p$ it becomes a $p$-dg category. We prove that the defining relations of the Hecke algebra are satisfied in the $p$-dg Grothendieck group. We conjecture that the $p$-dg Grothendieck group is isomorphic to the Iwahori-Hecke algebra, equipping it with a basis which may differ from both the Kazhdan-Lusztig basis and the $p$-canonical basis. More precise conjectures will be found in the sequel. Here are some other results contained in this paper. We provide an incomplete proof of the classification of all degree $+2$ derivations on the diagrammatic Hecke category, and a complete proof of the classification of those derivations for which the defining relations of the Hecke algebra are satisfied in the $p$-dg Grothendieck group. In particular, our special derivation is unique up to duality and equivalence. We prove that no such derivation exists in simply-laced types outside of finite and affine type $A$. We also examine a particular Bott-Samelson bimodule in type $A_7$, which is indecomposable in characteristic $2$ but decomposable in all other characteristics. We prove that this Bott-Samelson bimodule admits no nontrivial fantastic filtrations in any characteristic, which is the analogue in the $p$-dg setting of being indecomposable.Comment: 44 pages, many figures, color viewing essential. V2 contains corrections from referee reports. To appear in Transactions of the AM

Thick Soergel calculus in type A

Let R be the polynomial ring in n variables, acted on by the symmetric group S_n. Soergel constructed a full monoidal subcategory of R-bimodules which categorifies the Hecke algebra, whose objects are now known as Soergel bimodules. Soergel bimodules can be described as summands of Bott-Samelson bimodules (attached to sequences of simple reflections), or as summands of generalized Bott-Samelson bimodules (attached to sequences of parabolic subgroups). A diagrammatic presentation of the category of Bott-Samelson bimodules was given by the author and Khovanov in previous work. In this paper, we extend it to a presentation of the category of generalized Bott-Samelson bimodules. We also diagrammatically categorify the representations of the Hecke algebra which are induced from trivial representations of parabolic subgroups. The main tool is an explicit description of the idempotent which picks out a generalized Bott-Samelson bimodule as a summand inside a Bott-Samelson bimodule. This description uses a detailed analysis of the reduced expression graph of the longest element of S_n, and the semi-orientation on this graph given by the higher Bruhat order of Manin and Schechtman.Comment: Changed title. Expanded the exposition of the main proof. This paper relies extensively on color figure

Graph cohomology and Kontsevich cycles

The dual Kontsevich cycles in the double dual of associative graph homology correspond to polynomials in the Miller-Morita-Mumford classes in the integral cohomology of mapping class groups. I explain how the coefficients of these polynomials can be computed using Stasheff polyhedra and results from my previous paper GT/0207042.Comment: 36 pages, 3 figure