发表机构
Izmir Institute of Technology（伊兹密尔理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究虚拟纽结链环在连通和下基于双拟群的不变量，证明了基本双拟群为推出，并给出计数矩阵与虚拟括号不变量的连通和公式。
AI 中文摘要
本文研究了虚拟纽结链环（virtual knotoids）在连通和（connected sum）下基于双拟群（biquandle）的不变量的行为。我们首先证明了两个虚拟纽结链环的连通和的基本双拟群（fundamental biquandle）是双拟群范畴中一个跨（span）的推出（pushout）。通过将 Hom 函子应用于该推出描述，我们得到了 $K_1\\# K_2$ 的双拟群着色与加数（summands）的相容着色对之间的对应关系。这为已知的连通和下双拟群计数矩阵的矩阵乘积公式提供了范畴论解释。随后，我们研究了双拟群虚拟括号不变量（biquandle virtual bracket invariants）在连通和下的行为。我们证明，对于连通和 $K_1\\#K_2$ 的每个着色，其对应于加数 $K_1$ 和 $K_2$ 的相容着色对，归一化的双拟群虚拟括号值等于加数归一化值的乘积。在此基础上，我们得到了利用双拟群虚拟括号定义的归一化多重集不变量（normalized multiset invariants）的连通和公式。当系数环为数环（number ring）时，归一化括号多重集可以用多项式和具有多项式条目的矩阵来编码。我们引入了单项式上的乘积 $\star$ 以及由此诱导的矩阵乘积 $\odot$。然后我们证明了归一化的双拟群虚拟括号矩阵满足 \\[ \widetilde{\mathcal{M}}_X^{\beta}(K_1\\#K_2) = \widetilde{\mathcal{M}}_X^{\beta}(K_1) \odot \widetilde{\mathcal{M}}_X^{\beta}(K_2). \\]
英文摘要
In this paper, we study the behavior of biquandle-based invariants of virtual knotoids under their connected sum. We first show that the fundamental biquandle of the connected sum of two virtual knotoids is the pushout of a span in the category of biquandles. By applying the Hom functor to this pushout description, we obtain the correspondence between biquandle colorings of $K_1\# K_2$ and compatible pairs of colorings of summands. This provides a categorical explanation of a known matrix product formula for biquandle counting matrices under connected sum. We then study the behavior of biquandle virtual bracket invariants under connected sum. We show that, for each coloring of the connected sum $K_1\#K_2$ corresponding to a compatible pair of colorings of the summands $K_1$ and $K_2$, the normalized biquandle virtual bracket value factors as the product of the normalized values of the summands. Building on this, we obtain connected-sum formulas for the normalized multiset invariants defined by utilizing biquandle virtual brackets. When the coefficient ring is a number ring, the normalized bracket multisets can be encoded by polynomials and matrices with polynomial entries. We introduce a product $\star$ on monomials and an induced matrix product $\odot$. We then show that the normalized biquandle virtual bracket matrices satisfy \[ \widetilde{\mathcal{M}}_X^β(K_1\#K_2) = \widetilde{\mathcal{M}}_X^β(K_1) \odot \widetilde{\mathcal{M}}_X^β(K_2). \]
Comments24 pages, 11 figures
