3,298 research outputs found
Dimers and cluster integrable systems
We show that the dimer model on a bipartite graph on a torus gives rise to a
quantum integrable system of special type - a cluster integrable system. The
phase space of the classical system contains, as an open dense subset, the
moduli space of line bundles with connections on the graph. The sum of
Hamiltonians is essentially the partition function of the dimer model. Any
graph on a torus gives rise to a bipartite graph on the torus. We show that the
phase space of the latter has a Lagrangian subvariety. We identify it with the
space parametrizing resistor networks on the original graph.We construct
several discrete quantum integrable systems.Comment: This is an updated version, 75 pages, which will appear in Ann. Sci.
EN
On the welded Tube map
This note investigates the so-called Tube map which connects welded knots,
that is a quotient of the virtual knot theory, to ribbon torus-knots, that is a
restricted notion of fillable knotted tori in the 4-sphere. It emphasizes the
fact that ribbon torus-knots with a given filling are in one-to-one
correspondence with welded knots before quotient under classical Reidemeister
moves and reformulates these moves and the known sources of non-injectivity of
the Tube map in terms of filling changes.Comment: 23 pages ; v2: an error corrected and stylistic modifications ; to
appear in Contemporary Mathematic
Scissor equivalence for torus links
This article is about a natural distance function induced by smooth
cobordisms between links. We show that the cobordism distance of torus links is
determined by the profiles of their signature functions, up to a constant
factor.Comment: 15 pages, 5 figures, Theorem 1 adde
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