[Paper Review] Cube number can detect chirality and Legendrian type of knots

Ben McCarty|arXiv (Cornell University)|Jun 24, 2010

Geometric and Algebraic Topology7 references3 citations

TL;DR

This paper introduces the cube number as a knot invariant that detects chirality in all tested 7-crossing knots and distinguishes Legendrian types of certain torus knots, particularly (5,2)-torus knots with different rotation numbers. It demonstrates that the Legendrian cube number, defined as the minimal size of a cube diagram projecting to a Legendrian front, can differentiate between Legendrian isotopy classes even when standard invariants fail, using combinatorial moves on grid diagrams to prove minimality.

ABSTRACT

For a knot K the cube number is a knot invariant defined to be the smallest n for which there is a cube diagram of size n for K. We will show that the cube number detects chirality in all cases computed thus far, and distinguishes certain legendrian knots.

Motivation & Objective

• To establish the cube number as a new invariant capable of detecting chirality in knots.
• To define and investigate the Legendrian cube number as a potential invariant for distinguishing Legendrian knot types.
• To explore whether cube diagrams can detect differences in Legendrian isotopy classes despite lacking natural geometric compatibility with Legendrian structures.
• To verify the conjecture that for (p,2)-torus knots, Legendrian cube number distinguishes maximal and minimal rotation number types.
• To use computational and combinatorial methods to analyze minimal grid diagrams and their lifts to cube diagrams.

Proposed method

• Define a cube diagram as a 3D lattice embedding in an n×n×n grid where projections to each coordinate plane are grid diagrams with specific marking and crossing conditions.
• Use X-bend decompositions of grid diagrams to lift to z-cube bends in cube diagrams, ensuring orientation compatibility.
• Apply cyclic permutations and commutation moves to transform grid diagrams on a torus, preserving knot type and enabling minimality analysis.
• Establish that cube number equals arc index for certain knots, using known minimal grid diagrams as starting points.
• Use computer verification to count distinct minimal grid diagrams for Legendrian knots, particularly for (5,2)-torus knots with r = -3 and r = 3.
• Prove that if a minimal grid diagram contains unavoidable Type 1 configurations under all cyclic permutations, then its Legendrian cube number exceeds the arc index.

Experimental results

Research questions

• RQ1 Can the cube number detect chirality in all chiral knots, particularly those with up to 7 crossings?
• RQ2 Does the Legendrian cube number distinguish between Legendrian isotopy classes of (p,2)-torus knots with different rotation numbers?
• RQ3 Are there Legendrian knots with the same topological type and arc index but different Legendrian cube numbers?
• RQ4 Can the standard grid diagram of a (p,2)-torus knot be reached from any minimal diagram via commutation and cyclic permutation moves?
• RQ5 Is there a systematic relationship between the presence of Type 1 configurations in grid diagrams and the minimality of the corresponding cube diagram?

Key findings

• The cube number detects chirality in all eight of the first twelve chiral knots in Rolfsen’s table, with no counterexamples found in the remaining four.
• For the (5,2)-torus knot, the Legendrian cube number distinguishes between the left-handed Legendrian type with r = -3 and r = 3, with cℓ(Kmin) > α(T5,2) and cℓ(Kmax) = α(T5,2).
• A computer program confirmed that only 49 distinct minimal grid diagrams exist for the (5,2)-torus knot with r = -3, all reachable via cyclic permutations, and all retain at least one Type 1 configuration.
• The conjecture that cℓ(Kmax) < cℓ(Kmin) for (p,2)-torus knots with maximal and minimal rotation numbers is proven under the assumption of Conjecture 5.2.
• For the (7,2)-torus knot, cube number distinguishes Legendrian types with tb=0 and r=±5, even though both have the same arc index.
• The study shows that cube number is not always sufficient to detect Legendrian type, as some (p,2)-torus knots with same arc index and topological type still have equal cube numbers despite differing Legendrian invariants.