Twisted Virtual Bikeigebras and Twisted Virtual Handlebody-Knots

We generalize unoriented handlebody-links to the twisted virtual case, obtaining Reidemeister moves for handlebody-links in ambient spaces of the form $\Sigma\times [0,1]$ for $\Sigma$ a compact closed 2-manifold up to stable equivalence. We introduce a related algebraic structure known as twisted virtual bikeigebras whose axioms are motivated by the twisted virtual handlebody-link Reidemeister moves. We use twisted virtual bikeigebras to define $X$-colorability for twisted virtual handlebody-links and define an integer-valued invariant $\Phi_{X}^{\mathbb{Z}}$ of twisted virtual handlebody-links. We provide example computations of the new invariants and use them to distinguish some twisted virtual handlebody-links.