• foams
• geometric topology
• link homology
• Quantum invariants

The aim of this thesis is to give an overview of the symmetric Khovanov-Rozansky link homologies introduced by L.H. Robert and E. Wagner, and later focus on the symmetric gl(1)-homology after the paper by L. Marino.

The symmetric Khovanov-Rozansky link homologies were first introduced as a categorification of the Reshetikhin-Turaev invariants associated to symmetric powers of the standard representation of Uq(sl(N)). One of the methods used to compute the sl(N) quantum invariants is the planar graphic calculus developed by Murakami, Ohtsuki and Yamada, referred to as exterior MOY calculus. MOY calculus gives a combinatorial way to describe the coloured link invariants. Starting from the works of Queffelec-Rose and Queffelec-Rose-Sartori, Robert and Wagner used an approach similar to that used in the exterior case to categorify the symmetric MOY calculus. In this thesis we concentrate specifically on the case N=1, where the use of decorated vinyl graphs gives a more concrete description of the symmetric gl(1)-homology.

We first introduce the source category Foam, used to define the categorifying functor in the exterior case. In this category objects are MOY graphs while morphisms are given by foams. We define the exterior evaluation of foams and recall the universal construction used to define the functor F(N).

In order to apply the same construction in the symmetric case one has to restrict the source category. Therefore we consider the category with objects vinyl graphs and morphisms vinyl foams. Starting from this new category we define the functor S(N) via a universal construction, which categorifies the symmetric MOY calculus. We then obtain the symmetric link homologies by considering the hyper-rectangle of resolutions corresponding to a fixed link and applying the functor S(N) to it.

Finally, we restrict to the case N=1. We define decorated vinyl graphs, i.e vinyl graphs together with a choice of a family of homogeneous symmetric polynomials, and work with them instead of foams. In this case we are able to define a basis given by specific graphs called DUR graphs (decoration upper right), and use it to construct an algorithm to compute the invariants for uncoloured links. We conclude by describing the algorithm together with a running example.