Local transformations and functorial maps

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 $Δ$-equivalence of tangles in a fixed surface is classified by the extended homotopy index polynomial.

• The paper establishes a rigorous framework for picture-valued invariants by formalizing local transformation rules and crossing traits in knot diagrams.
• It distinguishes derivations from functorial maps and introduces explicit constructions like order and lifting maps applicable to classical and virtual knot theories.
• The study unifies diverse knot theories by employing categorical tools to classify sibling and flattable knots, offering new methods for knot equivalence.

Local Transformations and Functorial Maps: A Technical Overview

Introduction

The work "Local transformations and functorial maps" (2301.12478) provides a comprehensive and formal framework for picture-valued invariants derived from local transformation rules in knot and tangle diagrammatics. The author establishes a taxonomy and universal description of these invariants, associated with traits (notably parities, indices, and their generalizations) on diagram crossings. The paper rigorously distinguishes derivations and functorial maps—two classes of local transformations—and presents novel mechanisms for constructing such invariants, with applications to virtual and classical knot theory, as well as to extensions such as sibling and flattable knots.

Foundations: Traits, Diagram Categories, and Local Moves

The notion of a trait is axiomatized as an assignment of elements from a coefficient set (e.g., $\mathbb{Z}_2$ for parity, or more general abelian groups for indices) to the crossings of a diagram, compatibly with the diagrammatic moves that define the knot theory under study. Diagram categories are built from sets of diagrams and local moves (including and extending various forms of Reidemeister, virtual, flanking, smoothing, and forbidden moves). In this categorical setting, the universal trait is shown to exist and is characterized via signed indices, incorporating (for knots in a surface) the component index, order index, and homotopy class data.

These structures enable a uniform treatment of classical, virtual, flat, and free knot theories, as well as their regular versions (those omitting certain moves), and support the construction of skein modules and diagram categories as appropriate for various target spaces of invariants.

Classification of Picture-Valued Invariants

Picture-valued invariants—those whose values are sums or collections of diagrams of a certain type—are formalized in terms of two distinct operations:

• Derivations: For a given trait $\tau$, the derivation $d_\tau(D) = \sum_{c\in\mathcal{C}(D)} R_\tau(D, c)$ applies the local transformation to each crossing in the diagram $D$ individually and sums the results. Classical examples include the Turaev cobracket, Goldman's bracket, and Henrich's glueing invariant.
• Functorial Maps: The functorial map $f_\tau(D)$ applies the local rule to all crossings of $D$ simultaneously, producing a new diagram or a linear combination of diagrams. The Kauffman bracket, parity bracket, and parity projection fit within this paradigm.

The paper provides rigorous, sufficient conditions for the well-definedness and invariance of these operations under the relevant knot theory moves, including detailed compatibility requirements for the target theory.

Binary Functorial Maps and Sibling Knots

Among the significant technical contributions is the analysis of binary functorial maps, where the chosen schematic distinguishes crossings based on the values of a binary trait (e.g., parity, index, or order). The author presents new examples:

• Order Functorial Map: Parametrized by a (pre)order on the surface group's elements, this map gives rise to the concept of sibling knots: knots such that any diagram of one can be converted to a diagram of the other via a sequence of crossing switches determined by the order trait. The structure and realization of orders is detailed via compatibility conditions on homotopy data at crossings, leading to an algebraic description in terms of surface group orderings and preorders.
• Lifting Map: This functorial map inverts the "forgetful" map from virtual (or classical) knots to flat knots by endowing flat diagrams with under/over data according to a suitable order, effectively "lifting" the flat knot back to a virtual knot. The precise conditions under which a flat knot is liftable (i.e., results in a genuine knot under this procedure) are tied to the existence of discrete left-invariant orderings on the relevant surface group.

The explicit classification of liftable/flattenable knots is provided, with the result that liftability is tied to homotopic nontriviality and the absence of proper powers in the surface group element representing the knot.

Skein Modules, Extended Homotopy Index, and Generalization

An appendix supplies explicit computations and universal presentations of skein modules resulting from smoothing operations in various diagrammatics settings, including both oriented and unoriented cases. The paper generalizes and codifies skein-theoretic statements, culminating in a characterization theorem: $\Delta$-equivalence classes of tangles in a fixed surface are completely classified by the extended homotopy index polynomial, a complete invariant in the presence of $\Delta$-moves. This simultaneously unifies earlier results for classical linking numbers and their extensions to broader settings.

Numerical and Structural Results

Several explicit invariant formulas are established, for instance:

• The explicit formula for the Kauffman bracket extension in the presence of binary traits.
• The transformation rules for constructing skein module elements from diagrams via smoothing or crossing change rules, including the normalization and cyclotomic conditions arising from the surface's Euler characteristic.
• The precise identification of sibling knots, with concrete examples drawn from Green's table of virtual knots.

Implications and Future Directions

This work systematizes and unifies a vast class of picture-valued invariants via categorical and algebraic techniques, highlighting the central role of crossing traits (parities, indices, and orders) in their construction and analysis. The clarity in distinguishing derivations and functorial maps, coupled with the universal descriptions in terms of traits and local rules, offers a template for generating new invariants and probing the structure of knot and tangle categories well beyond the classical scope.

On a theoretical level, the analysis deepens connections between knot theory, orderability of surface groups, and combinatorial topology. Practically, new invariants such as the order functorial map may be used to distinguish knot types traditionally elusive to polynomial or Vassiliev-type invariants, and sibling knots represent a subtle equivalence notion not captured by classical invariants.

The paper raises several technical questions:

• The extension of functorial maps beyond the binary case (towards higher multi-valued traits).
• The full characterization of sibling and kindred knots, and whether essential concordance and slicing properties persist across sibling relationships.
• The interplay between the structure of functorial maps and algebraic structures, such as biquandles and orderable groups.

Conclusion

"Local transformations and functorial maps" formalizes and extends the theory of picture-valued invariants in knot theory, providing new conceptual and technical tools. The systematic framework for local transformations, grounded in the theory of crossing traits and functorial operations, not only encompasses known invariants but also enables the construction and classification of new classes of invariants and knot relationships—such as sibling and flattable knots—providing fertile ground for further algebraic, topological, and combinatorial exploration.