Oliver Pechenik

CLASSICAL AND VIRTUAL PSEUDODIAGRAM THEORY AND NEW BOUNDS ON UNKNOTTING NUMBERS AND GENUS

A pseudodiagram is a diagram of a knot with some crossing information missing. We review and expand the theory of pseudodiagrams introduced by Hanaki. We then extend this theory to the realm of virtual knots, a generalization of knots. In particular, we analyze the trivializing number of a pseudodiagram, i.e. the minimum number of crossings that must be resolved to produce the unknot. We consider how much crossing information is needed in a virtual pseudodiagram to identify a non-trivial knot, a classical knot, or a non-classical knot. We then apply pseudodiagram theory to develop new upper bounds on unknotting number, virtual unknotting number, and genus.

Proofs and generalizations of a homomesy conjecture of Propp and Roby

Let G be a group acting on a set X of combinatorial objects, with finite orbits, and consider a statistic ? : X ? C . Propp and Roby defined the triple ( X , G , ? ) to be homomesic if for any orbits O 1 , O 2 , the average value of the statistic ? is the same, that is 1 | O 1 | ? x ? O 1 ? ( x ) = 1 | O 2 | ? y ? O 2 ? ( y ) . In 2013 Propp and Roby conjectured the following instance of homomesy. Let SSY T k ( m × n ) denote the set of semistandard Young tableaux of shape m × n with entries bounded by k . Let S be any set of boxes in the m × n rectangle fixed under 180? rotation. For T ? SSY T k ( m × n ) , define ? S ( T ) to be the sum of the entries of T in the boxes of S . Let { P } be a cyclic group of order k where P acts on SSY T k ( m × n ) by promotion. Then ( SSY T k ( m × n ) , { P } , ? S ) is homomesic.We prove this conjecture, as well as a generalization to cominuscule posets. We also discuss analogous questions for tableaux with strictly increasing rows and columns under the K-promotion of Thomas and Yong, and prove limited results in that direction.

Genomic tableaux

We explain how genomic tableaux [Pechenik–Yong ’15] are a semistandard complement to increasing tableaux [Thomas–Yong ’09]. From this perspective, one inherits genomic versions of jeu de taquin, Knuth equivalence, infusion and Bender–Knuth involutions, as well as Schur functions from (shifted) semistandard Young tableaux theory. These are applied to obtain new Littlewood–Richardson rules for K-theory Schubert calculus of Grassmannians (after [Buch ’02]) and maximal orthogonal Grassmannians (after [Clifford–Thomas–Yong ’14], [Buch–Ravikumar ’12]). For the unsolved case of Lagrangian Grassmannians, sharp upper and lower bounds using genomic tableaux are conjectured.

Resonance in orbits of plane partitions and increasing tableaux

Abstract We introduce a new concept of resonance on discrete dynamical systems. This concept formalizes the observation that, in various combinatorially-natural cyclic group actions, orbit cardinalities are all multiples of divisors of a fundamental frequency. Our main result is an equivariant bijection between plane partitions in a box (or order ideals in the product of three chains) under rowmotion and increasing tableaux under K -promotion. Both of these actions were observed to have orbit sizes that were small multiples of divisors of an expected orbit size, and we show this is an instance of resonance, as K -promotion cyclically rotates the set of labels appearing in the increasing tableaux. We extract a number of corollaries from this equivariant bijection, including a strengthening of a theorem of Cameron and Fon-der-Flaass (1995) [9] and several new results on the order of K -promotion. Along the way, we adapt the proof of the conjugacy of promotion and rowmotion from Striker and Williams (2012) [38] to give a generalization in the setting of n -dimensional lattice projections. Finally we discuss known and conjectured examples of resonance relating to alternating sign matrices and fully-packed loop configurations.

A Midsummer Knot's Dream.

Summary We introduce playing games on the shadows of knots and demonstrate two novel games, namely, “To Knot or Not to Knot” and “Much Ado about Knotting.” We discuss winning strategies for these games on certain families of knot shadows and go on to suggest variations of these games for further study.

GENERALIZED GRAPH CORDIALITY

Hovey introduced A-cordial labelings in [4] as a simultaneous generalization of cordial and harmonious labelings. If A is an abelian group, then a labeling f : V (G) → A of the vertices of some graph G induces an edgelabeling on G; the edge uv receives the label f(u) + f(v). A graph G is A-cordial if there is a vertex-labeling such that (1) the vertex label classes differ in size by at most one and (2) the induced edge label classes differ in size by at most one. Research on A-cordiality has focused on the case where A is cyclic. In this paper, we investigate V4-cordiality of many families of graphs, namely complete bipartite graphs, paths, cycles, ladders, prisms, and hypercubes. We find that all complete bipartite graphs are V4-cordial except Km,n where m,n ≡ 2(mod 4). All paths are V4-cordial except P4 and P5. All cycles are V4-cordial except C4, C5, and Ck, where k ≡ 2(mod 4). All ladders P2�Pk are V4-cordial except C4. All prisms are V4-cordial except P2�Ck, where k ≡ 2(mod 4). All hypercubes are V4-cordial, except C4. Finally, we introduce a generalization of A-cordiality involving digraphs and quasigroups, and we show that there are infinitely many Q-cordial digraphs for every quasigroup Q.

Rhombic tilings and Bott–Samelson varieties

S.~Elnitsky (1997) gave an elegant bijection between rhombic tilings of

Equivariant

2n

K

-gons and commutation classes of reduced words in the symmetric group on

-theory of Grassmannians

n