The aim of this talk is to present some homology theories for graphs (and, in particular, multipath homology), describe their mutual relationship, and see how these theories are connected with other homology theories (i.e. homology theories for knots and algebras).
We start by describing some classical theories for graphs. Then, we will focus on the construction, due to Helme-Guizon and Rong, of chromatic homology. This theory categorifies the chromatic polynomial of graphs, and its construction is based on Khovanov homology. In particular, we will present a result due to J. Przytycki, which relates Khovanov homology, chromatic homology, and Hochschild homology. Afterwards, we will describe Turner and Wagner's approach to the "extension" of chromatic homology to oriented graphs while preserving an analogue of Przytycki's result. Finally, we will define multipath homology and compare it with Turner-Wagner homology.
In the last part of the talk, time permitting, we will give a topological description of the path poset, which is a key ingredient in the definition of Turner-Wagner and multipath homologies.
This talk is based on joint work with L. Caputi (University of Aberdeen) and S. Di Trani (Sapienza Università di Roma).
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