Parity in knot theory

Parity Bracket Polynomial via Some Parity of Virtual Links

“…Remark 1. For virtual knots, the parity given in Definition 2.1 is the same as the Gaussian parity of virtual knots defined by Manturov [5].…”

“…We may assume that the number of classical crossings of D is less than or equal to that of D . If D(D ) has one component, it has been proven by Manturov [5]. Thus we assume that D(D ) has at least two components.…”

“…A virtual link is the equivalence class of such a link diagram by generalized Reidemeister moves, which consist of (classical) Reidemeister moves of type R 1 , R 2 and R 3 and virtual Reidemeister moves of type V R 1 , V R 2 , V R 3 and the semivirtual move V R 4 as shown in Figure 1. Manturov [5] investigated parity properties of virtual knots so that every real crossing is declared to be even or odd according to a certain rule. In particular, he strengthens known invariants such as the Kauffman polynomial for virtual knots by using parities of real crossings so that he introduces the parity bracket polynomial for virtual knots.…”

Virtual Crossing Numbers for Virtual Knots

“…In this way, in [7] I proved that free knots are generally not invertible: this was done by means of finding a good non-invertible representative for free links and some other orientation-sensitive parity arguments. …”

“…The study of parity has been first undertaken in [7], see also [8,14] where functorial mappings from virtual knots to virtual knots were constructed, minimality theorems were proved, and many virtual knot invariants were refined. In the paper [15], by using parity, I constructed a diagrammatic projection mapping from virtual knots to classical knots.…”