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Abstract graphs associated to link diagrams
by
Lorenzo Traldi
Lafayette College
Invariants derived from a knot or link diagram are usually defined by considering geometric properties of the diagram in the plane; a classical example is the checkerboard graph, which is defined using the diagram's complementary regions. There is another combinatorial structure associated to a diagram though, namely an abstract 2-in, 2-out directed graph, equipped with two partitions: its vertices are partitioned into those that correspond to positive crossings and those that correspond to negative crossings; and its edge-set is partitioned into the circuits that correspond naturally to the link components. Here "abstract" refers to a graph that is given without any preferred imbedding in the plane (or any other surface). The same construction applies to virtual link diagrams; the digraph corresponding to a virtual link diagram has vertices corresponding to the classical crossings.
There are several advantages to thinking of a diagram this way. One is that the combinatorial structure is simple: there are no complementary regions, cyclic orders of incident edges at vertices, under/overpassing arcs etc. Also, circuit partitions in 2-in, 2-out digraphs have received a great deal of attention from graph theorists in recent decades, so there are good combinatorial techniques available. Finally, abstract graphs represent classical and virtual knots and links with no necessary distinction between the two, just as abstract graph theory incorporates planar and non-planar graphs without distinguishing between them. For instance, abstract graphs have Reidemeister moves corresponding to the classical ones, but no Reidemeister moves corresponding to the virtual ones: they aren't needed because there is simply no information to which they might refer.
We have been able to verify that the Jones polynomial and Kauffman bracket can be derived from these abstract graphs. Identifying other invariants that can be derived from them is an open problem.
Paper reference: arXiv:0901.1451
Date received: January 27, 2009
Copyright © 2009 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cayk-02.