Biquandle-Based Invariants of Virtual Knotoids and Connected Sum
Abstract
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 K1 # K2 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 introduce normalized multiset invariants derived from biquandle virtual brackets and show that they are strictly stronger than their unnormalized versions in certain cases by an explicit example. Furthermore, we show that for each coloring of the connected sum K1 # K2 corresponding to a compatible pair of colorings of the summands K1 and K2, 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 an induced matrix product ⊙ and show that the normalized biquandle virtual bracket matrices satisfy M̃Xβ(K1 # K2) = M̃Xβ(K1) ⊙ M̃Xβ(K2).
Related
BibTeX
@misc{kayaslan2026biquandlebased,
title = {{Biquandle-Based Invariants of Virtual Knotoids and Connected Sum}},
author = {Hamdi Kayaslan and Selçuk İlbeyli},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07187},
eprint = {2610.07187},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07187},
}
RIS
TY - UNPB
TI - Biquandle-Based Invariants of Virtual Knotoids and Connected Sum
AU - Hamdi Kayaslan
AU - Selçuk İlbeyli
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.07187
UR - https://arxiv.org/abs/2610.07187
AB - 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 introduce normalized multiset invariants derived from biquandle virtual brackets and show that they are strictly stronger than their unnormalized versions in certain cases by an explicit example. Furthermore, 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 an induced matrix product ⊙ and show that the normalized biquandle virtual bracket matrices satisfy M̃_X^β(K_1 # K_2) = M̃_X^β(K_1) ⊙ M̃_X^β(K_2).
ER -