[Paper Review] Bikei Invariants and Gauss Diagrams for Virtual Knotted Surfaces

Sam Nelson, Patricia Rivera|arXiv (Cornell University)|Sep 27, 2014

Geometric and Algebraic Topology11 references3 citations

TL;DR

This paper extends bikei counting invariants to virtual knotted surfaces using Gauss diagrams, providing a combinatorial method to characterize orientability. It introduces Gauss diagrams for marked vertex diagrams, proves that orientability is equivalent to an even number of chord endpoints between each chord's endpoints, and shows bikei invariants are well-defined for all virtual marked vertex diagrams via involutory biquandles (bikei).

ABSTRACT

Marked vertex diagrams provide a combinatorial way to represent knotted surfaces in $\mathbb{R}^4$; including virtual crossings allows for a theory of virtual knotted surfaces and virtual cobordisms. Biquandle counting invariants are defined only for marked vertex diagrams representing knotted orientable surfaces; we extend these invariants to all virtual marked vertex diagrams by considering colorings by involutory biquandles, also known as bikei. We introduce a way of representing marked vertex diagrams with Gauss diagrams and use these to characterize orientability.

Motivation & Objective

• To extend bikei counting invariants to all virtual marked vertex diagrams, including non-orientable surfaces, by using involutory biquandles (bikei).
• To introduce Gauss diagrams for marked vertex diagrams as a computer-friendly representation for virtual knotted surfaces.
• To characterize orientability of virtual knotted surfaces combinatorially using Gauss diagrams and endpoint parity conditions.
• To establish that bikei invariants are well-defined for all virtual marked vertex diagrams, overcoming limitations of standard biquandle invariants.
• To investigate the relationship between orientability and 2-colorability in virtual knotted surfaces, showing they do not always coincide.

Proposed method

• Represent virtual knotted surfaces using marked vertex diagrams (ch-diagrams), which encode knotted surfaces in R⁴ with classical and virtual crossings.
• Define Gauss diagrams for marked vertex diagrams by embedding the diagram on a circle, labeling classical and saddle crossings, and using directed chords for classical crossings and marked chords for saddle crossings.
• Use the chord endpoint count between endpoints of each chord as a criterion for orientability: a diagram is orientable iff every chord has an even number of endpoints between its two ends.
• Apply bikei (involuntary biquandle) colorings to virtual marked vertex diagrams, ensuring invariance under Yoshikawa moves.
• Construct biquandle matrices to encode bikei operations, enabling algorithmic computation of bikei counting invariants.
• Prove that global orientation can be consistently assigned if and only if the endpoint parity condition is satisfied, linking topological orientability to a combinatorial invariant.

Experimental results

Research questions

• RQ1 Can bikei counting invariants be extended to all virtual marked vertex diagrams, including non-orientable surfaces?
• RQ2 Is there a combinatorial characterization of orientability for virtual knotted surfaces using Gauss diagrams?
• RQ3 Does orientability of a virtual knotted surface correspond to 2-colorability, and if not, what is the precise relationship?
• RQ4 What enhancements to the bikei counting invariant are possible in the virtual setting, especially with nontrivial virtual crossing operations?
• RQ5 How do virtual crossings affect the structure of bikei invariants and the topological invariants of knotted surfaces?

Key findings

• The bikei counting invariant is well-defined for all virtual marked vertex diagrams, including non-orientable surfaces, due to the use of involutory biquandles (bikei).
• A marked vertex Gauss diagram is orientable if and only if, for every chord, the number of chord endpoints between its two endpoints is even.
• For unknotted orientable surfaces of genus g, the bikei counting invariant equals |X|, the size of the bikei X.
• For unknotted non-orientable surfaces with cross-cap number c, the bikei counting invariant equals |F|, where F = {x ∈ X | xˣ = xₓ = x} is the set of fixed points under the bikei operations.
• The 2-coloring invariant (with X = ℤ₂, xʸ = xₓ = x+1) yields Φ_X^ℤ = 2 for all unknotted orientable surfaces and Φ_X^ℤ = 0 for all unknotted non-orientable surfaces.
• Orientability and 2-colorability do not coincide in the virtual knotted surface setting, as demonstrated by a diagram that is 2-colorable but not orientable.