One of the most important development of the last 15 years was
Khovanov's categorification of the Jones polynomial of links. In
parallel, the theory of homology of racks and quandles and its
applications to knot theory was developed. We propose and develop
here an idea how to connect distributive (e.g. rack or quandle)
homology and Khovanov homology, via Yang-Baxter operators (the
fundamental tool of statistical physics). As a special case of my
ideas (technical but easy to visualize), I will show how to
construct a \(q\)-polynomial for rooted trees (or more generally
finite plane poset). This becomes a tool to analyze the Kauffman
bracket skein modules of \(3\)-manifolds (starting from a cable
crossing). 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 coefficient \(q^{r(v_i)}\), 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. We prove several properties of \(Q(T)\), e.g. that it is
tree invariant, that is, does not depend on a plane embedding. We
relate \(Q(T)\) to Gauss polynomials (\(q\)-binomial coefficients). The
application of \(Q(T)\) to a lattice crossing will be presented in a
join paper with M. Dąbkowski and C. Li.
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