2009/01/11 by Lorenzo Traldi, Traldi, Lorenzo
Computer Science · Mathematics · #05C50 #57M25 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Topological and Geometric Data Analysis #math.CO #math.GT #msc:05C50 #msc:57M25
paper · pdf · doi:10.48550/arxiv.0901.1451
43 pages, 14 figures. Revisions may be made before publication in the Journal of Knot Theory and its Ramifications
openalex publication_date 2009/01/11 · arxiv created 2009/03/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let D be an oriented classical or virtual link diagram with directed universe U. Let C denote a set of directed Euler circuits, one in each connected component of U. There is then an associated looped interlacement graph L(D,C) whose construction involves very little geometric information about the way D is drawn in the plane; consequently L(D,C) is different from other combinatorial structures associated with classical link diagrams, like the checkerboard graph, which can be difficult to extend to arbitrary virtual links. L(D,C) is determined by three things: the structure of U as a 2-in, 2-out digraph, the distinction between crossings that make a positive contribution to the writhe and those that make a negative contribution, and the relationship between C and the directed circuits in U arising from the link components; this relationship is indicated by marking the vertices where C does not follow the incident link component(s). We introduce a bracket polynomial for arbitrary marked graphs, defined using either a formula involving matrix nullities or a recursion involving the local complement and pivot operations; the marked-graph bracket of L(D,C) is the same as the Kauffman bracket of D. This provides a unified combinatorial description of the Jones polynomial that applies seamlessly to both classical and non-classical virtual links.