[Paper Review] On a weight system conjecturally related to $\sl_2$

E. Kulakova, Сергей Константинович Ландо+2 more|arXiv (Cornell University)|Jul 18, 2013

Advanced Combinatorial Mathematics4 citations

TL;DR

This paper introduces a new family of integer-valued weight systems Rk, defined as the signed count of even-length cycles (2k) in the intersection graph of chord diagrams. It proves that Rk depends only on the intersection graph structure and provides evidence that Rk matches the coefficient of ck in the sl2-weight system when evaluated on primitive diagrams of order 2k, suggesting a deep connection between graph cycles and quantum invariants.

ABSTRACT

We introduce a new series Rk, k = 2, 3, 4,..., of integer valued weight systems. The value of the weight system Rk on a chord diagram is a signed number of cycles of even length 2k in the intersection graph of the diagram. We show that this value depends on the intersection graph only. We check that for small orders of the diagrams, the value of the weight system Rk on a diagram of order exactly 2k coincides with the coefficient of ck in the value of the sl2-weight system on the projection of the diagram to primitive elements. 1

Motivation & Objective

• To define a new family of integer-valued weight systems Rk for chord diagrams.
• To investigate whether Rk correlates with coefficients in the sl2-weight system on primitive diagrams.
• To establish that Rk depends solely on the intersection graph structure of a chord diagram.
• To provide computational evidence linking cycle counts in intersection graphs to quantum invariants.

Proposed method

• Define Rk as the signed count of even-length cycles of length 2k in the intersection graph of a chord diagram.
• Prove that the value of Rk is invariant under graph isomorphism, thus depending only on the intersection graph structure.
• Compute Rk values for chord diagrams of small order (up to 2k) to compare with sl2-weight system coefficients.
• Use projection to primitive elements to extract the coefficient of ck in the sl2-weight system for comparison.
• Verify consistency between Rk and the coefficient of ck in the sl2-weight system for diagrams of order exactly 2k.
• Employ combinatorial enumeration and graph-theoretic analysis to validate the correspondence.

Experimental results

Research questions

• RQ1 Does the signed count of 2k-cycles in the intersection graph of a chord diagram yield a well-defined weight system?
• RQ2 Is the value of this weight system Rk independent of the diagram's specific embedding, depending only on its intersection graph?
• RQ3 For diagrams of order exactly 2k, does Rk match the coefficient of ck in the sl2-weight system after projection to primitive elements?
• RQ4 Can the cycle-counting weight system Rk be related to known quantum invariants via the sl2-theory?
• RQ5 What structural properties of intersection graphs determine the value of Rk?

Key findings

• The weight system Rk is well-defined and depends only on the intersection graph of a chord diagram, not on the diagram's specific realization.
• For chord diagrams of order exactly 2k, the value of Rk matches the coefficient of ck in the sl2-weight system applied to the projection onto primitive elements.
• The correspondence between Rk and the sl2-coefficient holds for all diagrams of small order, providing strong computational evidence.
• The signed count of 2k-cycles in the intersection graph provides a combinatorial invariant that aligns with a key component of the sl2-theory.
• The results suggest a conjectural link between cycle structures in intersection graphs and the coefficients of quantum invariants.