Biquandle-Based Invariants of Virtual Knotoids under Connected Sum

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 $K_1\mathrm{\#}{K}_2$ 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 $K_1\mathrm{\#}{K}_2$ 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 $${\tilde{\mathcal{M}}}_X^{\beta }(K_1\mathrm{\#}{K}_2)={\tilde{\mathcal{M}}}_X^{\beta }(K_1)\odot {\tilde{\mathcal{M}}}_X^{\beta }(K_2)\mathrm{.}$$