A note on Grid Homology in lens spaces: Z\mathbb{Z} coefficients and computations

We present a combinatorial proof for the existence of the sign refined Grid Homology in lens spaces, and a self contained proof that $\partial_\mathbb{Z}^2 = 0$. We also present a Sage program that computes $\widehat{GH} (L(p,q),K;\mathbb{Z})$, and provide empirical evidence supporting the absence of torsion in these groups.Comment: 27 pages, 23 figure

Pattern Avoidance and the Bruhat Order

The structure of order ideals in the Bruhat order for the symmetric group is elucidated via permutation patterns. A method for determining non-isomorphic principal order ideals is described and applied for small lengths. The permutations with boolean principal order ideals are characterized. These form an order ideal which is a simplicial poset, and its rank generating function is computed. Moreover, the permutations whose principal order ideals have a form related to boolean posets are also completely described. It is determined when the set of permutations avoiding a particular set of patterns is an order ideal, and the rank generating functions of these ideals are computed. Finally, the Bruhat order in types B and D is studied, and the elements with boolean principal order ideals are characterized and enumerated by length.Comment: 18 pages, 7 figure

The area of cyclic polygons: Recent progress on Robbins' Conjectures

In his works [R1,R2] David Robbins proposed several interrelated conjectures on the area of the polygons inscribed in a circle as an algebraic function of its sides. Most recently, these conjectures have been established in the course of several independent investigations. In this note we give an informal outline of these developments.Comment: To appear in Advances Applied Math. (special issue in memory of David Robbins

Conjugacy in Garside Groups III: Periodic braids

An element in Artin's braid group B_n is said to be periodic if some power of it lies in the center of B_n. In this paper we prove that all previously known algorithms for solving the conjugacy search problem in B_n are exponential in the braid index n for the special case of periodic braids. We overcome this difficulty by putting to work several known isomorphisms between Garside structures in the braid group B_n and other Garside groups. This allows us to obtain a polynomial solution to the original problem in the spirit of the previously known algorithms. This paper is the third in a series of papers by the same authors about the conjugacy problem in Garside groups. They have a unified goal: the development of a polynomial algorithm for the conjugacy decision and search problems in B_n, which generalizes to other Garside groups whenever possible. It is our hope that the methods introduced here will allow the generalization of the results in this paper to all Artin-Tits groups of spherical type.Comment: 33 pages, 13 figures. Classical references implying Corollaries 12 and 15 have been added. To appear in Journal of Algebr

Spindle configurations of skew lines

We prove a conjecture of Crapo and Penne which characterizes isotopy classes of skew configurations with spindle-structure. We use this result in order to define an invariant, spindle-genus, for spindle-configurations. We also slightly simplify the exposition of some known invariants for configurations of skew lines and use them to define a natural partition of the lines in a skew configuration. Finally, we describe an algorithm which constructs a spindle in a given switching class, or proves non-existence of such a spindle.Comment: 42 pages, many figures. A new corrected proof of a conjecture of Crapo and Penne is added. More new material is also adde

BFACF-style algorithms for polygons in the body-centered and face-centered cubic lattices

In this paper the elementary moves of the BFACF-algorithm for lattice polygons are generalised to elementary moves of BFACF-style algorithms for lattice polygons in the body-centred (BCC) and face-centred (FCC) cubic lattices. We prove that the ergodicity classes of these new elementary moves coincide with the knot types of unrooted polygons in the BCC and FCC lattices and so expand a similar result for the cubic lattice. Implementations of these algorithms for knotted polygons using the GAS algorithm produce estimates of the minimal length of knotted polygons in the BCC and FCC lattices