3,477 research outputs found
Singular link Floer homology
We define a grid presentation for singular links i.e. links with a finite
number of rigid transverse double points. Then we use it to generalize link
Floer homology to singular links. Besides the consistency of its definition, we
prove that this homology is acyclic under some conditions which naturally make
its Euler characteristic vanish.Comment: 29 pages, many figure
Zonal polynomials via Stanley's coordinates and free cumulants
We study zonal characters which are defined as suitably normalized
coefficients in the expansion of zonal polynomials in terms of power-sum
symmetric functions. We show that the zonal characters, just like the
characters of the symmetric groups, admit a nice combinatorial description in
terms of Stanley's multirectangular coordinates of Young diagrams. We also
study the analogue of Kerov polynomials, namely we express the zonal characters
as polynomials in free cumulants and we give an explicit combinatorial
interpretation of their coefficients. In this way, we prove two recent
conjectures of Lassalle for Jack polynomials in the special case of zonal
polynomials.Comment: 45 pages, second version, important change
On k-Convex Polygons
We introduce a notion of -convexity and explore polygons in the plane that
have this property. Polygons which are \mbox{-convex} can be triangulated
with fast yet simple algorithms. However, recognizing them in general is a
3SUM-hard problem. We give a characterization of \mbox{-convex} polygons, a
particularly interesting class, and show how to recognize them in \mbox{} time. A description of their shape is given as well, which leads to
Erd\H{o}s-Szekeres type results regarding subconfigurations of their vertex
sets. Finally, we introduce the concept of generalized geometric permutations,
and show that their number can be exponential in the number of
\mbox{-convex} objects considered.Comment: 23 pages, 19 figure
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