Narrow Results

We find and study properties of the coefficients $C_i^{(m,n)}(A)$ and ${\overline{C}}_j^{(m,n)}(A)$ of Catalan states with the maximal number of returns and no nesting in the Kauffman bracket sum of an $m\times n$ lattice crossing $L(m,n)$. We show that, up to a power of $A$, $C_i^{(m,n)}(A)$ and ${\overline{C}}_j^{(m,n)}(A)$ are polynomials in variable $A^4$ with unimodal and palindromic coefficients. Our results can be viewed as a step toward obtaining closed form formulas for coefficients of an arbitrary Catalan state obtained from $L(m,n)$.

We show which crossingless connections between $2(m+n)$ outer boundary points of an annulus can be realized as Kauffman states of the B-type lattice crossing $L_{\mathrm{\text{B}}}(m,n)$. Furthermore, we give a closed-form formula for the number of realizable B-type Catalan states, and we find coefficients of those obtained as Kauffman states of $L_{\mathrm{\text{B}}}(m,1)$ and $L_{\mathrm{\text{B}}}(m,2)$.

Plucking polynomial of a plane rooted tree with a delay function $\alpha$ was introduced in 2014 by Przytycki. As shown in this paper, plucking polynomial factors when $\alpha$ satisfies additional conditions. We use this result and ${\mathrm{\Theta }}_A$-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 give closed-form formulas for coefficients of Catalan states of $L(m,3)$.