On the axioms of singquandles

Abstract

In this paper we deal with the notion of singquandles introduced in [CEHN17]. This is an algebraic structure that naturally axiomatizes Reidemeister moves for singular links, similarly to what happens for ordinary links and quandle structure. We present a new axiomatization that shows different algebraic aspects and simplifies applications. We also reformulate and simplify the axioms for affine singquandles (in particular in the idempotent case).

On the axioms of singquandles

“…The paper is organized as follows: in the first section we recall the basics about binary algebraic structures and right quasigroups and quandles in particular. The second one is about Oriented singquandles, and we resume the main results of [BC21]. In the next two sections we deal with stuquandles and oriented bondles and we show that such structured can be build from oriented singquandles.…”

“…The same approach has been adapted to virtual knots and the related Reidemeister moves and so oriented singquandles have been defined [CCE20]. These structures can be studied in a purely algebraic way and an axiomatization of them (alternative to the original one) can be found in [BC21].…”