[Paper Review] On the Alexander polynomial of lens space knot

Motoo Tange|arXiv (Cornell University)|Sep 24, 2014

Geometric and Algebraic Topology3 references3 citations

TL;DR

This paper refines Ozsváth-Szabó's result on the Alexander polynomial of lens space knots by introducing a geometric curve model in R² that encodes the distribution of non-zero coefficients. The curve's structure reveals a new invariant, the α-index, which constrains the polynomial's form and enables classification of lens space knots with specific Alexander polynomials, including those matching (2,r)-torus knots and low-genus examples.

ABSTRACT

Ozsváth-Szabó proved the property that any coefficient of Alexander polynomial of lens space knot is either $\pm1$ or $0$ and the non-zero coefficients are alternating. Combining the formulas of the Alexander polynomial of lens space knots due to Kadokami-Yamada and Ichihara-Saito-Teragaito, we refine Ozsváth-Szabó's property as the existence of simple curves included in a region in ${\Bbb R}^2$. The existence of curves, that has no end-points connected, is just 1-component in a region, can search distribution of non-zero coefficients of the Alexander polynomial of the lens space knot. This curve is much useful to obtain constraints of Alexander polynomials of lens space knots. For example, we can investigate the location of the second, third and fourth non-zero coefficients. The curve extracts new invariant $α$-index. The invariant is an important factor to determine Alexander polynomial of lens space knot. We classify lens space surgeries that the Alexander polynomial is the same as a $(2,r)$-torus knot and lens space surgeries with small genus and so on. As well as lens space knots in $S^3$, we also deal with lens space knots in homology spheres, which the surgery duals are simple (1,1)-knots.

Motivation & Objective

• To refine Ozsváth-Szabó's property that coefficients of lens space knot Alexander polynomials are ±1 or 0 and alternate in sign.
• To develop a geometric method using simple curves in R² to visualize and constrain the distribution of non-zero coefficients in the Alexander polynomial.
• To define and utilize a new invariant, the α-index, to classify lens space knots with specific Alexander polynomials.
• To classify lens space knots whose Alexander polynomials match those of (2,r)-torus knots and other low-genus examples.
• To extend the analysis to lens space knots in homology spheres, particularly those with surgery duals that are simple (1,1)-knots.

Proposed method

• Constructs a curve in R² whose connected components correspond to non-zero coefficient positions in the Alexander polynomial of a lens space knot.
• Uses the curve's topological structure—specifically, the absence of endpoints and single-component nature—to derive constraints on coefficient distribution.
• Applies formulas for Alexander polynomials of lens space knots from Kadokami-Yamada and Ichihara-Saito-Teragaito to express ΔK(t) in a symmetrized form.
• Defines the α-index as a new invariant derived from the curve's geometry, which helps determine the Alexander polynomial's structure.
• Employs the pillowcase method to compute and list lens space knots Kp,k in non-L-space homology spheres up to genus 30.
• Uses the dA-function and A-function sequences to analyze coefficient patterns and apply Lemma 4.5 to rule out certain configurations.

Experimental results

Research questions

• RQ1 What geometric structure in R² can encode the distribution of non-zero coefficients in the Alexander polynomial of a lens space knot?
• RQ2 How does the α-index, derived from this geometric structure, constrain the form of the Alexander polynomial?
• RQ3 Which lens space knots have Alexander polynomials identical to those of (2,r)-torus knots?
• RQ4 What are the possible genus and coefficient sequences for lens space knots with small genus, and how can they be classified?
• RQ5 Can the curve model and α-index be used to classify lens space knots in non-L-space homology spheres?

Key findings

• The existence of a simple, closed curve in R² with no endpoints and one connected component corresponds exactly to the non-zero coefficient positions in the Alexander polynomial of a lens space knot.
• The α-index, derived from the curve's geometry, is a crucial invariant that determines the Alexander polynomial's structure and enables classification of lens space knots.
• For genus g ≤ 5, the only lens space knots with half non-zero sequence (g, g−1, g−3, g−4, ..., 1, 0) are the (3,4)-torus knot (g=3) and the (−2,3,7)-pretzel knot (g=5).
• The Alexander polynomial of the (3,4)-torus knot is ΔT(3,4)(t) = t^4 − t^3 + t^2 − t + 1 − t^{−1} + t^{−2} − t^{−3} + t^{−4}, with surgery parameter (19,7,8).
• The (−2,3,7)-pretzel knot has Alexander polynomial ΔPr(−2,3,7)(t) = t^6 − t^5 + t^4 − t^3 + t^2 − t + 1 − t^{−1} + t^{−2} − t^{−3} + t^{−4} − t^{−5} + t^{−6}, with surgery parameter (11,3,4).
• Tables 2 and 3 list 28 lens space knots Kp,k in non-L-space homology spheres with genus up to 30, including their parameters (p,k,k₂), non-zero coefficient sequences, and α-index values.