17,601 research outputs found
Alexander invariants of ribbon tangles and planar algebras
Ribbon tangles are proper embeddings of tori and cylinders in the
-ball~, "bounding" -manifolds with only ribbon disks as
singularities. We construct an Alexander invariant of ribbon
tangles equipped with a representation of the fundamental group of their
exterior in a free abelian group . This invariant induces a functor in a
certain category of tangles, which restricts to the exterior
powers of Burau-Gassner representation for ribbon braids, that are analogous to
usual braids in this context. We define a circuit algebra over
the operad of smooth cobordisms, inspired by diagrammatic planar algebras
introduced by Jones, and prove that the invariant commutes with
the compositions in this algebra. On the other hand, ribbon tangles admit
diagrammatic representations, throught welded diagrams. We give a simple
combinatorial description of and of the algebra ,
and observe that our construction is a topological incarnation of the Alexander
invariant of Archibald. When restricted to diagrams without virtual crossings,
provides a purely local description of the usual Alexander
poynomial of links, and extends the construction by Bigelow, Cattabriga and the
second author
The self-dual point of the two-dimensional random-cluster model is critical for
We prove a long-standing conjecture on random-cluster models, namely that the
critical point for such models with parameter on the square lattice is
equal to the self-dual point . This gives a
proof that the critical temperature of the -state Potts model is equal to
for all . We further prove that the transition is
sharp, meaning that there is exponential decay of correlations in the
sub-critical phase. The techniques of this paper are rigorous and valid for all
, in contrast to earlier methods valid only for certain given . The
proof extends to the triangular and the hexagonal lattices as well.Comment: 27 pages, 10 figure
On the scaling limits of planar percolation
We prove Tsirelson's conjecture that any scaling limit of the critical planar
percolation is a black noise. Our theorems apply to a number of percolation
models, including site percolation on the triangular grid and any subsequential
scaling limit of bond percolation on the square grid. We also suggest a natural
construction for the scaling limit of planar percolation, and more generally of
any discrete planar model describing connectivity properties.Comment: With an Appendix by Christophe Garban. Published in at
http://dx.doi.org/10.1214/11-AOP659 the Annals of Probability
(http://www.imstat.org/aop/) by the Institute of Mathematical Statistics
(http://www.imstat.org
- …