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On Virtual Crossing Number Estimates For Virtual Links. arXiv:0811.0712
"... We address the question of detecting minimal virtual diagrams with respect to the number of virtual crossings. This problem is closely connected to the problem of detecting the minimal number of additional intersection points for a generic immersion of a singular link in R 2. We tackle this problem ..."
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We address the question of detecting minimal virtual diagrams with respect to the number of virtual crossings. This problem is closely connected to the problem of detecting the minimal number of additional intersection points for a generic immersion of a singular link in R 2. We tackle this problem by the socalled ξpolynomial whose leading (lowest) degree naturally estimates the virtual crossing number. Several sufficient conditions for minimality together with infinite series of new examples are given. We also state several open questions about Mdiagrams, which are minimal according to our sufficient conditions. 1
A bracket polynomial for graphs. II. Links, Euler circuits and marked graphs
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"... 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 t ..."
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Cited by 1 (1 self)
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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 2in, 2out 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. If the number of connected components in U is c(U), the Kauffman bracket [D] and the markedgraph bracket [L(D, C)] are related by [D] = d c(U)−1 · [L(D, C)]. This provides a unified combinatorial description of the Jones polynomial that applies seamlessly to both classical and nonclassical virtual links.