A bracket polynomial for graphs. II. Links, Euler circuits and marked graphs
| dc.creator | Traldi, Lorenzo | |
| dc.date | 2009-01-11 | |
| dc.date | 2009-03-04 | |
| dc.date.accessioned | 2026-07-07T12:48:29Z | |
| dc.date.available | 2026-07-07T12:48:29Z | |
| dc.description | Let $D$ be an oriented classical or virtual link diagram with directed universe $\vec{U}$. 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 $\vec{U}$ 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 $\vec{U}$ 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. | |
| dc.description | 43 pages, 14 figures. Revisions may be made before publication in the Journal of Knot Theory and its Ramifications | |
| dc.identifier | https://arxiv.org/abs/0901.1451 | |
| dc.identifier | http://arxiv.org/abs/0901.1451 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/222070 | |
| dc.subject | Geometric Topology | |
| dc.subject | Combinatorics | |
| dc.subject | 57M25, 05C50 | |
| dc.title | A bracket polynomial for graphs. II. Links, Euler circuits and marked graphs | |
| dc.type | text |