6 research outputs found
Christoffel words and the strong Fox conjecture for two-bridge knots
The trapezoidal Fox conjecture states that the coefficient sequence of the
Alexander polynomial of an alternating knot is unimodal. We are motivated by a
harder question, the strong Fox conjecture, which asks whether the coefficient
sequence of the Alexander polynomial of alternating knots is actually
log-concave. Our approach is to introduce a polynomial associated
to a Christoffel word and to prove that its coefficient sequence is
log-concave. This implies the strong Fox conjecture for two-bridge knots.Comment: 19 pages; 9 figure
Symmetries and intrinsic vs. extrinsic properties of
We consider the following question: How much of the combinatorial structure
determining properties of is ``intrinsic'' and
how much new information do we obtain from using properties specific to this
space? Our approach is to study the effect of the -action. Apart from
being a natural action to consider, it is known that this action does not
extend to other wonderful compactifications associated to the
hyperplane arrangement. We find the differences in intersection patterns of
faces on associahedra and permutohedra which characterize the failure to extend
to other compactifications and show that this is reflected by most terms of
degree of the cohomology/Chow ring.
Even from a combinatorial perspective, terms of degree 1 are more naturally
related to geometric properties. In particular, imposing -invariance
implies that many of the log concave sequences obtained from degree 1
Hodge--Riemann relations (and all of them for ) on the Chow ring of
can be restricted to those with a special
recursive structure. A conjectural result implies that this is true for all
. Elements of these sequences can be expressed as polynomials in quantum
Littlewood--Richardson coefficients multiplied by terms such as partition
components, factorials, and multinomial coefficients. After dividing by
binomial coefficients, polynomials with these numbers as coefficients can be
interepreted in terms of volumes or resultants. Finally, we find a connection
between the geometry of and higher degree
Hodge--Riemann relations of other rings via Toeplitz matrices.Comment: 19 pages, Comments welcome
The extremals of the Kahn-Saks inequality
A classical result of Kahn and Saks states that given any partially ordered
set with two distinguished elements, the number of linear extensions in which
the ranks of the distinguished elements differ by is log-concave as a
function of . The log-concave sequences that can arise in this manner prove
to exhibit a much richer structure, however, than is evident from log-concavity
alone. The main result of this paper is a complete characterization of the
extremals of the Kahn-Saks inequality: we obtain a detailed combinatorial
understanding of where and what kind of geometric progressions can appear in
these log-concave sequences. This settles a partial conjecture of
Chan-Pak-Panova, while the analysis uncovers new extremals that were not
previously conjectured. The proof relies on a much more general geometric
mechanism -- a hard Lefschetz theorem for nef classes that was obtained in the
setting of convex polytopes by Shenfeld and Van Handel -- which forms a model
for the investigation of such structures in other combinatorial problems.Comment: 29 pages, 1 figur