Coefficients of Catalan states of lattice crossing II: Applications of ΘA-state expansions

IF 0.3 4区数学 Q4 MATHEMATICS Journal of Knot Theory and Its Ramifications Pub Date : 2024-04-18 DOI:10.1142/s0218216524500032

Mieczyslaw K. Dabkowski, Cheyu Wu

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Abstract

Plucking polynomial of a plane rooted tree with a delay function $α$ was introduced in 2014 by 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 give closed-form formulas for coefficients of Catalan states of $L (m, 3)$ .

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晶格交叉的加泰罗尼亚态系数 II： ΘA 态展开的应用

Przytycki 于 2014 年提出了具有延迟函数 α 的平面有根树的拔取多项式。正如本文所示，当 α 满足附加条件时，拔取多项式会产生因子。我们利用这一结果和之前工作中引入的 ΘA 态扩展，推导出 (m×n)- 格子交叉 L(m,n) 所产生的加泰罗尼亚态 C 的系数 C(A) 的新特性。特别是，我们证明了当 C 具有具有某些特殊性质的弧时，C(A) 的系数。在许多情况下，这将为计算 C(A) 提供更有效的方法。作为应用，我们给出了 L(m,3) 的加泰罗尼亚态系数的闭式公式。

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来源期刊

Journal of Knot Theory and Its Ramifications 数学-数学

CiteScore

0.80

自引率

40.00%

发文量

127

审稿时长

4-8 weeks

期刊介绍： This Journal is intended as a forum for new developments in knot theory, particularly developments that create connections between knot theory and other aspects of mathematics and natural science. Our stance is interdisciplinary due to the nature of the subject. Knot theory as a core mathematical discipline is subject to many forms of generalization (virtual knots and links, higher-dimensional knots, knots and links in other manifolds, non-spherical knots, recursive systems analogous to knotting). Knots live in a wider mathematical framework (classification of three and higher dimensional manifolds, statistical mechanics and quantum theory, quantum groups, combinatorics of Gauss codes, combinatorics, algorithms and computational complexity, category theory and categorification of topological and algebraic structures, algebraic topology, topological quantum field theories). Papers that will be published include: -new research in the theory of knots and links, and their applications; -new research in related fields; -tutorial and review papers. With this Journal, we hope to serve well researchers in knot theory and related areas of topology, researchers using knot theory in their work, and scientists interested in becoming informed about current work in the theory of knots and its ramifications.

期刊最新文献

Computation of the knot symmetric quandle and its application to the plat index of surface-links Coefficients of Catalan states of lattice crossing II: Applications of ΘA-state expansions Knot quandle decomposition along a torus Heegaard Floer invariants for cyclic 3-orbifolds Presentations of diquandles and diquandle coloring invariants for solid torus knots and links