[Paper Review] Alexander Polynomial Invariants of Torus Knots T(n,3) and Chebyshev Polynomials

A. M. Gavrilik, A.M. Pavlyuk|arXiv (Cornell University)|Jul 27, 2011

Geometric and Algebraic Topology4 citations

TL;DR

This paper derives an explicit formula expressing the Alexander polynomial Δₙ,₃(t) of torus knots T(n,3) as a finite sum of Alexander polynomials Δₖ,₂(t) of torus knots T(k,2), and further expresses Δₙ,₃(t) in terms of Chebyshev polynomials of the first and second kind. The key contribution extends this connection to general torus knots T(n,l) with n and l coprime, showing that Δₙ,ₗ(t) depends on n through Chebyshev polynomials with arguments involving Δₙ,₂(t) or Δₙ,₃(t).

ABSTRACT

The explicit formula, which expresses the Alexander polynomials \\Delta_{n,3}(t) of torus knots T(n,3) as a sum of the Alexander polynomials \\Delta_{k,2}(t) of torus knots T(k,2), is found. Using this result and those from our previous papers, we express the Alexander polynomials \\Delta_{n,3}(t) through Chebyshev polynomials. The latter result is extended to general torus knots T(n,l) with n and l coprime.

Motivation & Objective

• To derive an explicit formula for the Alexander polynomial Δₙ,₃(t) of torus knots T(n,3) in terms of Δₖ,₂(t) for various k.
• To establish a connection between Δₙ,₃(t) and Chebyshev polynomials of the first and second kind.
• To generalize the result to arbitrary torus knots T(n,l) with n and l coprime, showing dependence on n through Chebyshev polynomials.
• To demonstrate that the functional dependence of Δₙ,ₗ(t) on n is fully encoded in Δₙ,₂(t) or Δₙ,₃(t), depending on coprimality conditions.
• To provide a framework for expressing knot invariants via algebraic structures like q-numbers and Chebyshev polynomials, with potential applications in field theory.

Proposed method

• Use of the skein relation and normalization condition to define Alexander polynomials Δₙ,ₗ(t) for torus knots.
• Application of q-number formalism and recurrence relations for Chebyshev polynomials Tₙ(x) and Vₙ(x) to derive identities.
• Derivation of the identity Δₙ,₃(t) = Σ cₖ Δₖ,₂(t) for specific coefficients cₖ, linking T(n,3) to T(k,2) knots.
• Expression of Δₙ,ₗ(t) via Chebyshev polynomials: Δₙ,ₗ(t) = Vₗ₋₁(Tₙ(y)) / Vₗ₋₁(y), where y = t¹ᐟ² + t⁻¹ᐟ².
• Establishment of alternative forms: Δₙ,ₗ(t) = Vₙ₋₁(Tₗ(y)) / Vₙ₋₁(y), highlighting symmetric dependence on n and l.
• Use of variable substitution z = (t¹ᐟ² + t⁻¹ᐟ²)Δₙ,₂(t) to express Δₙ,ₗ(t) as a function of Δₙ,₂(t), showing that all n-dependence is captured in Δₙ,₂(t).

Experimental results

Research questions

• RQ1 Can the Alexander polynomial Δₙ,₃(t) of torus knots T(n,3) be expressed as a finite sum of Δₖ,₂(t) polynomials for appropriate k?
• RQ2 How are the Alexander polynomials Δₙ,₃(t) related to Chebyshev polynomials of the first and second kind?
• RQ3 Can the dependence of Δₙ,ₗ(t) on n be fully characterized through Δₙ,₂(t) or Δₙ,₃(t), given coprimality conditions?
• RQ4 What is the general algebraic structure linking Δₙ,ₗ(t) to Chebyshev polynomials for arbitrary coprime n and l?
• RQ5 Is there a unified expression for Δₙ,ₗ(t) that separates the roles of n and l via Tₙ(x) and Vₗ₋₁(x) respectively?

Key findings

• The Alexander polynomial Δₙ,₃(t) for torus knots T(n,3) is expressed as a finite sum of Δₖ,₂(t) polynomials, with coefficients determined by recurrence and q-number identities.
• Δₙ,₃(t) is explicitly expressed in terms of Chebyshev polynomials via Δₙ,₃(t) = V₂(Tₙ(y)) / V₂(y), where y = t¹ᐟ² + t⁻¹ᐟ².
• For general torus knots T(n,l) with n and l coprime, Δₙ,ₗ(t) = Vₗ₋₁(Tₙ(y)) / Vₗ₋₁(y), showing that the n-dependence is encoded in Tₙ(y), while l-dependence is in the degree of the second-kind polynomial.
• The dependence of Δₙ,ₗ(t) on n is fully captured by Δₙ,₂(t) when n, l, and 2 are coprime, or by Δₙ,₃(t) when n, l, and 3 are coprime.
• The expression Δₙ,ₗ(t) = Vₙ₋₁(Tₗ(y)) / Vₙ₋₁(y) provides a symmetric dual representation, highlighting the duality between n and l in the polynomial structure.
• The results extend previous work on T(n,2) knots by showing that the connection between Alexander polynomials and Chebyshev polynomials holds universally for all coprime torus knots T(n,l).