[Paper Review] Some computational results on mod 2 finite-type invariants of knots and string links

Ted Stanford|ArXiv.org|May 27, 2004

Geometric and Algebraic Topology1 references3 citations

TL;DR

This paper presents a comprehensive table of primitive mod-2 finite-type invariants of order ≤6 for all prime knots with up to ten crossings, computed via an optimized implementation of Vassiliev's algorithm. It identifies key mod-2 congruences, including a novel chirality criterion derived from the Alexander polynomial via the invariant $v_{4a}$, and establishes that a mod-2 weight system of order 5 for 2-strand string links does not integrate to a finite-type invariant of order 5.

ABSTRACT

We publish a table of primitive finite-type invariants of order less than or equal to six, for knots of ten or fewer crossings. We note certain mod-2 congruences, one of which leads to a chirality criterion in the Alexander polynomial. We state a computational result on mod-2 finite-type invariants of 2-strand string links.

Motivation & Objective

• To compute and tabulate primitive finite-type invariants of order ≤6 for all prime knots with ≤10 crossings.
• To identify and analyze mod-2 congruences among invariants, particularly those linked to knot chirality.
• To extend the algorithm for finite-type invariants to 2-strand string links and investigate integrability of mod-2 weight systems.
• To provide a basis for invariants satisfying desirable algebraic and topological properties, including mirror symmetry behavior.
• To demonstrate that a mod-2 weight system of order 5 for 2-strand string links does not integrate to a finite-type invariant of order 5.

Proposed method

• An implementation of Vassiliev’s algorithm was used to compute invariants, with basis optimization to minimize integer values and enforce mod-2 congruences.
• The basis was chosen to satisfy mirror symmetry: even-order invariants are preserved under mirror image, odd-order invariants change sign.
• Mod-2 congruences were enforced by aligning values of invariants $v_{3}$ with $v_{4a}$, $v_{5a}$ with $v_{6a}$, etc., modulo 2.
• The algorithm was extended to compute finite-type invariants for 2-strand string links using the same framework.
• A computational check confirmed that a mod-2 weight system of order 5 for 2-strand string links does not integrate to a finite-type invariant of order 5.
• The results were cross-verified using known knot polynomials and translation matrices from Kanenobu’s notation.

Experimental results

Research questions

• RQ1 Can a mod-2 weight system of order 5 for 2-strand string links be integrated into a finite-type invariant of order 5?
• RQ2 What mod-2 congruences exist among primitive finite-type invariants of order ≤6 for knots with ≤10 crossings?
• RQ3 Does the invariant $v_{4a} = \frac{1}{2}(3a_2 - a_2^2) + a_4$ provide a chirality criterion independent of the knot determinant?
• RQ4 How can a basis for primitive invariants be chosen to satisfy both integrality and mirror symmetry properties simultaneously?
• RQ5 What is the structure of the sublattice of integer vectors realized by knot invariants under the chosen basis?

Key findings

• The paper provides a complete table of primitive finite-type invariants of order ≤6 for all 10-crossing and fewer prime knots, with values minimized to a maximum absolute value of 39.
• A chirality criterion is established: if $v_{4a}(K) \equiv 1 \pmod{2}$, then $K$ is chiral, and this condition is independent of the knot determinant.
• The invariant $v_{4a}$ modulo 2 is shown to be independent of the determinant, as there exist knots with the same determinant but differing $v_{4a}$ mod 2.
• A mod-2 weight system of order 5 for 2-strand string links was found to not integrate to a finite-type invariant of order 5, confirming a non-integrability result.
• The lattice of realized invariant values has index 16 in $\mathbb{Z}^{12}$, reflecting four essential mod-2 congruences among the basis invariants.
• The basis satisfies all desired properties except full integrality, with the invariant values forming a sublattice of $\mathbb{Z}^{12}$ of index 16 due to mod-2 constraints.