Parity in knot theory and graph-links

Abstract

The present monograph is devoted to low-dimensional topology in the context of two thriving theories: parity theory and theory of graph-links, the latter being an important generalization of virtual knot theory constructed by means of intersection graphs. Parity theory discovered by the second-named author leads to a new perspective in virtual knot theory, the theory of cobordisms in two-dimensional surfaces, and other new domains of topology. Theory of graph-links highlights a new combinatorial approach to knot theory.

Weak Parities and Functorial Maps

“…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

“…Crossings are called even (odd) respectively to the chords. The parity allows one to realise the following principle [4][5][6][9][10][11][12][13][14][15][16][17][18][19]: If a knot diagram is complicated enough then it realises itself. The latter means that it appears as a subdiagram in any diagram equivalent to it.…”

“…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.…”

A parity map of framed chord diagrams

“…In topology and graph theory, many notions often have their "odd", "non-orientable", "framed" counterparts, see for example [6,10,11,13,17,18,19,20,21,22,23,24]. Usually, even objects are better understood, however, in the odd case, it is much easier to catch the non-trivial information.…”