Biquandles With Structures Related to Virtual Links and Twisted Links

Abstract

We introduce two kinds of structures, called v-structures and t-structures, on biquandles. These structures are used for colorings of diagrams of virtual links and twisted links such that the numbers of colorings are invariants. Given a biquandle or a quandle, we give a method of constructing a biquandle with these structures. Using the numbers of colorings, we show that Bourgoin's twofoil and non-orientable virtual m-foils do not represent virtual links.

Twisted virtual biracks and their twisted virtual link invariants

“…Twisted virtual links were introduced in [1] and subsequently studied in works such as [7,9]. Twisted virtual links extend the concept of virtual links from previous work [10,8]; where virtual links arise by drawing link diagrams on compact orientable surfaces with nonzero genus, twisted virtual links arise when we draw link diagrams on compact surfaces allowing nonzero genus and nonzero cross-cap number.…”

“…In remark 2 we observed that trivial virtual and twisted operations do not generally give a birack the structure of a twisted virtual birack. In [9], a construction called the twisted product is given in which a birack B and a choice of automorphism of B are used to define a twisted virtual birack structure on the Cartesian product B × B. What other ways are there to define a twisted virtual birack given a birack?…”

“…Quandle-and biquandle-based invariants of twisted virtual links were first considered in [9]. In [14] a counting invariant of unframed oriented classical and virtual knots and links was defined using labelings by finite racks and extended to finite biracks in [15].…”

Virtual Shadow Modules and Their Link Invariants

“…If we also have (S • ∆) 1 = (S • ∆) 2 , X is a twisted virtual biquandle. See [9,17] for more. As with virtual biracks, we can represent a twisted virtual birack structure on a set X = {x 1 , .…”

“…Virtual knots and links were introduced in [18] and have been the subject of much study since. Twisted virtual knots and links were introduced in [4] and have been studied in papers such as [9,15,17]. Biracks (including biquandles) were first introduced in [13] as an algebraic structure defining invariants of framed knots and links in S 3 .…”