arXiv subjects
Publications and source records attributed to Cheyu Wu.
J. Hoste and J. H. Przytycki computed the Kauffman bracket skein module (KBSM) of lens spaces in their papers published in 1993 and 1995. Using a basis for the KBSM of a fibered torus, we construct new bases for the KBSMs of two families of lens spaces: $L(p,2)$ and $L(4k,2k+1)$ with $k\neq 0$. For KBSM of $L(0,1) = {\bf S}^{2}\times S^{1}$, we find a new generating set that yields its decomposition into a direct sum of cyclic modules.
We construct a family of bases for the Kauffman bracket skein module (KBSM) of the product of an annulus and a circle. Using these bases, we find a new basis for the KBSM of $(\beta,2)$-fibered torus as a first step toward developing techniques for computing KBSM of a family of small Seifert fibered $3$-manifolds.
Plucking polynomial of a plane rooted tree with a delay function $\alpha$ was introduced in 2014 by J.H.~Przytycki. As shown in this paper, plucking polynomial factors when $\alpha$ satisfies additional conditions. We use this result and $\Theta_{A}$-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 $L(m,3)$.
Plucking polynomial for plane rooted trees was introduced by J.H. Przytycki in 2014. As it was shown later, this polynomial can be used to find coefficients $C(A)$ of Catalan states $C$ of $m \times n$-lattice crossing $L(m,n)$ without returns on one side. In this paper, we show that $C(A)$ for any $C$ can be found by using $\Theta_{A}$-state expansion which represents $C(A)$ as a linear combination of coefficients of Catalan states with no top returns over $\mathbb{Q}(A)$. We also provide an algorithm for finding $\Theta_{A}$-state expansions and examples of its applications. Finally, an example of a Catalan state with non-unimodal coefficient is given.