Local transformations and functorial maps

TL;DR

This paper provides a comprehensive framework for picture-valued invariants in knot theory, introduces new functorial maps, and explores their applications to surface groups, virtual knots, and flat knots.

Contribution

It introduces two novel functorial maps, the order and lifting maps, and explores their implications for sibling knots and flat virtual knots.

Findings

Order functorial map relates to surface group orderings and sibling knots.

Lifting map connects virtual knots to flat knots, with examples of liftable flat knots.

Extended homotopy index polynomial classifies $ ext{Delta}$-equivalence of tangles.

Abstract

Picture-valued invariants are the main achievement of parity theory by V.O. Manturov. In the paper we give a general description of such invariants which can be assigned to a parity (in general, a trait) on diagram crossings. We distinguish two types of picture-valued invariants: derivations (Turaev bracker, index polynomial etc.) and functorial maps (Kauffman bracket, parity bracket, parity projection etc.). We consider some examples of binary functorial maps. Besides known cases of functorial maps, we present two new examples. The order functorial map is closely connected with (pre)orderings of surface groups and leads to the notion of sibling knots, i.e. knots such that any diagram of one knot can be transformed to a diagram of the other by crossing switching. The other is the lifting map which is inverse to forgetting of under-overcrossings information which turns virtual knots into flat knots. We give some examples of liftable flat knots and flattable virtual knots. An appendix of the paper contains description of some smoothing skein modules. In particular, we show that $\Delta$-equivalence of tangles in a fixed surface is classified by the extended homotopy index polynomial.