[Paper Review] Combinatorial Formulae for Finite-Type Invariants via Parities

Micah Chrisman, Vassily Olegovich Manturov|arXiv (Cornell University)|Feb 2, 2010

Geometric and Algebraic Topology19 references6 citations

TL;DR

This paper introduces new combinatorial formulae for virtual knot invariants using the concept of parity, which encodes embedding information not present in classical knots. These formulae yield invariants that are Kauffman finite-type but not of Goussarov-Polyak-Viro (GPV) finite-type, including an infinite family of integer-valued virtualization invariants of order n and eleven new non-trivial order-2 formulae outside the GPV framework.

ABSTRACT

The present paper produces examples of Gauss diagram formulae for virtual knot invariants which have no analogue in the classical knot case. These combinatorial formulae contain additional information about how a subdiagram is embedded in a virtual knot diagram. The additional information comes from the second author's recently discovered notion of parity. For a parity of flat virtual knots, the new combinatorial formulae are Kauffman finite-type invariants. However, many of the combinatorial formulae possess exotic properties. It is shown that there exists an integer valued virtualization invariant combinatorial formula of order n for every n (i.e. it is stable under the map which changes the direction of one arrow but preserves the sign). Hence, it is not of Goussarov-Polyak-Viro finite-type. Moreover, every homogeneous Polyak-Viro combinatorial formula admits a decomposition into an even part and an odd part. For the Gaussian parity, neither part of the formula is of GPV finite-type when it is nonconstant on the set of classical knots. In addition, eleven new non-trivial combinatorial formulae of order 2 are presented which are not of GPV finite-type.

Motivation & Objective

• To develop combinatorial formulae for virtual knot invariants that incorporate embedding data via parity, extending beyond classical knot theory.
• To identify and characterize invariants that are Kauffman finite-type but not of Goussarov-Polyak-Viro (GPV) finite-type.
• To demonstrate the existence of integer-valued virtualization invariants of every order n, challenging the GPV finite-type classification.
• To decompose homogeneous Polyak-Viro formulae into even and odd parts and analyze their GPV finite-type status.
• To present and validate eleven new non-trivial combinatorial formulae of order 2 that are not of GPV finite-type.

Proposed method

• Utilizes the second author’s recently introduced notion of parity for flat virtual knots to encode topological embedding information in Gauss diagrams.
• Applies the parity to construct new combinatorial formulae for virtual knot invariants that depend on subdiagram embedding.
• Employs the Gaussian parity to analyze the decomposition of formulae into even and odd parts and assesses their finite-type properties.
• Constructs an infinite family of integer-valued invariants stable under virtualization, showing they are not of GPV finite-type.
• Presents and verifies eleven new non-trivial order-2 formulae through combinatorial analysis, confirming their non-GPV finite-type nature.

Experimental results

Research questions

• RQ1 Can combinatorial formulae for virtual knot invariants be constructed using parity that encode embedding data absent in classical knots?
• RQ2 Are there virtual knot invariants of finite-type in the Kauffman sense that are not of Goussarov-Polyak-Viro finite-type?
• RQ3 Does every order-n virtualization invariant exist, and can such invariants be expressed as combinatorial formulae?
• RQ4 How do the even and odd parts of a homogeneous Polyak-Viro formula behave in terms of GPV finite-type when non-constant on classical knots?
• RQ5 Are there new, non-trivial combinatorial formulae of order 2 that lie outside the GPV finite-type framework?

Key findings

• An infinite family of integer-valued virtualization invariants of order n exists for every n, and these are not of GPV finite-type.
• For the Gaussian parity, neither the even nor the odd part of a non-constant homogeneous Polyak-Viro formula is of GPV finite-type when restricted to classical knots.
• Eleven new non-trivial combinatorial formulae of order 2 are constructed, all of which are not of GPV finite-type.
• The new formulae incorporate embedding information via parity, providing invariants that have no analogue in classical knot theory.
• The paper establishes that parity-based formulae can yield invariants stable under virtualization, a property incompatible with GPV finite-type classification.