Parity, free knots, groups, and invariants of finite type

Abstract

Abstract. In this paper, on the basis of the notion of parity introduced recently by the author, for each positive integer m we construct invariants of long virtual knots with values in some simply defined group G m ; conjugacy classes of this group play a role as invariants of compact virtual knots. By construction, each of the invariants is unaltered by the move of virtualization. Factorization of the group algebra of the corresponding group leads to invariants of finite order of (long) virtual knots that do not change under virtualization.The central notion used in the construction of the invariants is parity: the crossings of diagrams of free knots is endowed with an additional structure -each crossing is declared to be even or odd, where even crossings behave regularly under Reidemeister moves.

Weak Parities and Functorial Maps

“…The new invariants demonstrated the existence of nontrivial free knots as well as nontrivial cobordism classes of free knots [7].…”

“…Additional information, which is introduced into knot diagrams with parity, allows one to strengthen knot invariants [1,[5][6][7][8][9][10][11][12]. The new invariants demonstrated the existence of nontrivial free knots as well as nontrivial cobordism classes of free knots [7].…”

Picture-valued parity-biquandle bracket

“…Theorem 3.2 (cf. [4][5][6][9][10][11][12][13][14][15][16][17][18][19]). Let G be a leading term of [[L]](f ) j .…”

“…The bracket [·] is a diagram-valued invariant of (virtual, free) knots [4][5][6][9][10][11][12][13][14][15][16][17][18][19]. It is important for us to know that One of the simplest knot invariants is the colouring invariant: one colours edges of a knot diagram by colours from a given palette and counts some colourings which are called admissible.…”

“…In [3] we brought together the ideas from parity theory [4][5][6][9][10][11][12][13][14][15][16][17][18][19] and the ideas from [20] and constructed the universal biquandle picture-valued invariant of classical and virtual links. We axiomatised the new invariant in such a way that it a priori dominated both the biquandle bracket and the parity bracket, thus being a very strong invariant of both virtual and classical links: it is at least as strong as the biquandle invariant for classical knots and at least as strong as the parity bracket for virtual knots.…”