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Symmetric intersections of Rauzy fractals
In this article we study symmetric subsets of Rauzy fractals of unimodular
irreducible Pisot substitutions. The symmetry considered is reflection through
the origin. Given an unimodular irreducible Pisot substitution, we consider the
intersection of its Rauzy fractal with the Rauzy fractal of the reverse
substitution. This set is symmetric and it is obtained by the balanced pair
algorithm associated with both substitutions
Approximating Holant problems by winding
We give an FPRAS for Holant problems with parity constraints and
not-all-equal constraints, a generalisation of the problem of counting
sink-free-orientations. The approach combines a sampler for near-assignments of
"windable" functions -- using the cycle-unwinding canonical paths technique of
Jerrum and Sinclair -- with a bound on the weight of near-assignments. The
proof generalises to a larger class of Holant problems; we characterise this
class and show that it cannot be extended by expressibility reductions.
We then ask whether windability is equivalent to expressibility by matchings
circuits (an analogue of matchgates), and give a positive answer for functions
of arity three
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