Reduced alternating diagrams of links in thickened surfaces and of virtual links minimize crossing number, and any two such diagrams of the same link have the same writhe.
TG-Hyperbolicity of Virtual Links
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abstract
We extend the theory of hyperbolicity of links in the 3-sphere to tg-hyperbolicity of virtual links, using the fact that the theory of virtual links can be translated into the theory of links living in closed orientable thickened surfaces. When the boundary surfaces are taken to be totally geodesic, we obtain a tg-hyperbolic structure with a unique associated volume. We prove that all virtual alternating links are tg-hyperbolic. We further extend tg-hyperbolicity to several classes of non-alternating virtual links. We then consider bounds on volumes of virtual links and include a table for volumes of the 116 nontrivial virtual knots of four or fewer crossings, all of which, with the exception of the trefoil knot, turn out to be tg-hyperbolic.
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math.GT 1years
2019 1verdicts
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The Jones-Krushkal polynomial and minimal diagrams of surface links
Reduced alternating diagrams of links in thickened surfaces and of virtual links minimize crossing number, and any two such diagrams of the same link have the same writhe.