Rooted trees with the same plucking polynomial

Strict unimodality of q-polynomials of rooted trees

We classify rooted trees which have strictly unimodal q-polynomials (plucking polynomial). We also give criteria for a trapezoidal shape of a plucking polynomial. We generalize results of Pak and Panova on strict unimodality of q-binomial coefficients. We discuss which polynomials can be realized as plucking polynomials and whether or not different rooted trees can have the same plucking polynomial.

带延迟函数的拔除多项式的单峰性探索

Exploring unimodality of the plucking polynomial with delay function

本文研究带根树不变量拔除多项式及其带延迟函数形式，给出刺猬型带根树的公式并探索其单峰性，通过反单峰延迟函数和特定像集延迟函数提供例子与推测。

Catalan States of Lattice Crossing: Application of Plucking Polynomial

For a Catalan state $C$ of a lattice crossing $L\left( m,n\right) $ with no returns on one side, we find its coefficient $C\left( A\right) $ in the Relative Kauffman Bracket Skein Module expansion of $L\left( m,n\right) $. We show, in particular, that $C\left( A\right) $ can be found using the plucking polynomial of a rooted tree with a delay function associated to $C$. Furthermore, for $C$ with returns on one side only, we prove that $C\left( A\right) $ is a product of Gaussian polynomials, and its coefficients form a unimodal sequence.

Coefficients of Catalan States of Lattice Crossing II: Applications of $Θ_{A}$-state Expansions

Plucking polynomial of a plane rooted tree with a delay function $α$ was introduced in 2014 by J.H.~Przytycki. As shown in this paper, plucking polynomial factors when $α$ satisfies additional conditions. We use this result and $Θ_{A}$-state expansion introduced in our previous work to derive new properties of coefficients $C(A)$ of Catalan states $C$ resulting from an $m \times n$-lattice crossing $L(m,n)$. In particular, we show that $C(A)$ factors when $C$ has arcs with some special properties. In many instances, this yields a more efficient way for computing $C(A)$. As an application, we g…

Q-polynomial invariant of rooted trees

We describe in this note a new invariant of rooted trees. We argue that the invariant is interesting on it own, and that it has connections to knot theory and homological algebra. However, the real reason that we propose this invariant to readers is that we deal here with an elementary, interesting, new mathematics, and after reading this essay readers can take part in developing the topic, inventing new results and connections to other disciplines of mathematics, and likely, statistical mechanics, and combinatorial biology. We also provide a (free) translation of the paper in Polish.

Progress in distributive homology: from q-polynomial of rooted trees to Yang-Baxter homology

This is an extended abstract of the talk given at the Oberwolfach Workshop "Algebraic Structures in Low-Dimensional Topology", 25 May -- 31 May 2014. My goal was to describe progress in distributive homology from the previous Oberwolfach Workshop June 3 - June 9, 2012, in particular my work on Yang-Baxter homology; however I concentrated my talk on my recent discovery of q-polynomial of a rooted tree; the appropriate topic as my talk was on May 30, 2014, the 30 anniversary of the Jones polynomial, and the polynomial has its roots in the Kauffman bracket approach to the Jones polynomial.