Topology Atlas | Conferences

q-polynomial invariant of rooted trees
by
Jozef H. Przytycki
George Washington University and University of Maryland

It seems to be appropriate to propose, on this 30 anniversary of the Jones polynomial, yet another new, to my best knowledge, polynomial. It is a very simple polynomial of rooted trees but intimately connected to the Jones polynomial via the Kauffman bracket polynomial and the Temperley-Lieb algebra. Thus for a rooted tree (T, v) we construct the polynomial Q(T, v) ∈ Z[q]. Here is the definition, starting from plane rooted tree (later proved to be independent on plane embedding).

Definition: For a plane tree T with a root v we define a polynomial Q(T) by a recursive relation: Q(T) is equal to the sum over all leaves, v_i, of T of the polynomials Q(T-v_i) with the coefficients q^r(v_i), where r(v_i) is the number of edges of T to the right of v_i. We normalize Q(T) to be 1 at the root. Among many possible ways to define planar rooted tree polynomial, Q(T) has an advantage of being independent of plane embedding so it is rooted tree invariant. We give a formula for a root change and for a polynomial of a root product T=T₁∨T₂. We get: Q(T₁ ∨T₂) = ((E(T₁)+E(T₂)) || (E(T₁)))_qQ(T₁)(Q(T₂) Where (a || b)_q denotes a Gauss polynomial, that is q-binomial coefficient. For the use in knot theory we consider plane rooted tree with delay function f_T: E(T) → Z, where Q(T, f_T) is the sum only over those leaves with value function no more than 1, and f_T-v=f_T -1.

The application of Q(T) to a lattice crossing will be presented in a join paper with Mieczyslaw Dabkowski and Changsong Li.

In the broader context we will show (after Loday) that plane rooted trees form an almost simplicial set and we speculate about changing presimplicial modules into q-polynomials, by considering ∑_i=0ⁿqⁱd_i in place of ∑_i=0ⁿ(-1)ⁱd_i.

Date received: April 3, 2014

Copyright © 2014 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cbjb-02.