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The Axial Electric Potential and Length of a Torus Knot

Henry Jiang

TL;DR

The paper investigates the axial electric potential and field of a uniformly charged $(p,q)$-torus knot embedded on a torus, leveraging $z$-axis symmetry to reduce the problem to tractable one-dimensional integrals. It introduces a novel application of complex analysis and contour integration, including branch cuts, to both approximate the knot length and compute axial quantities, yielding explicit approximations and qualitative insights. A key finding is that the electric field along the $z$-axis vanishes only at the origin, while the axial potential is maximized at $\alpha=0$ and increases with knot winding $p/q$; the study shows how the length integral can be reduced to a segment integral and approximated analytically. Together, these methodological advances provide efficient, rigorous tools for analyzing specific torus knots in physical knot theory, with potential applications in materials science and molecular biology.

Abstract

Physical knot theory, where knots are treated like physical objects, is important to many fields. One natural problem is to give a knot a uniform charge, and analyze the resulting electric field and electric potential. There have been some results on the number of critical points of the electric potential from knots, such as by Lipton (2021) and Lipton, Townsend, and Strogatz (2022). However, little analysis has been done on the electric field and electric potential using calculations for specific knots. We focus on torus knots, specifically a parametrization that embeds it on a torus centered at the origin with rotational symmetry about the z-axis. Particularly, in this project, we analyze the electric field along the z-axis to take advantage of symmetry. We also analyze the length of the knot as a simpler integral. We show that the electric field is zero only at the origin, and investigate the extreme points of the electric field and electric potential using numerical methods and calculations. We also demonstrate a new way to apply methods for contour integration in complex analysis to calculate the length, electric potential, and electric field, and provide an explicit approximation for the length of a torus knot.

The Axial Electric Potential and Length of a Torus Knot

TL;DR

The paper investigates the axial electric potential and field of a uniformly charged -torus knot embedded on a torus, leveraging -axis symmetry to reduce the problem to tractable one-dimensional integrals. It introduces a novel application of complex analysis and contour integration, including branch cuts, to both approximate the knot length and compute axial quantities, yielding explicit approximations and qualitative insights. A key finding is that the electric field along the -axis vanishes only at the origin, while the axial potential is maximized at and increases with knot winding ; the study shows how the length integral can be reduced to a segment integral and approximated analytically. Together, these methodological advances provide efficient, rigorous tools for analyzing specific torus knots in physical knot theory, with potential applications in materials science and molecular biology.

Abstract

Physical knot theory, where knots are treated like physical objects, is important to many fields. One natural problem is to give a knot a uniform charge, and analyze the resulting electric field and electric potential. There have been some results on the number of critical points of the electric potential from knots, such as by Lipton (2021) and Lipton, Townsend, and Strogatz (2022). However, little analysis has been done on the electric field and electric potential using calculations for specific knots. We focus on torus knots, specifically a parametrization that embeds it on a torus centered at the origin with rotational symmetry about the z-axis. Particularly, in this project, we analyze the electric field along the z-axis to take advantage of symmetry. We also analyze the length of the knot as a simpler integral. We show that the electric field is zero only at the origin, and investigate the extreme points of the electric field and electric potential using numerical methods and calculations. We also demonstrate a new way to apply methods for contour integration in complex analysis to calculate the length, electric potential, and electric field, and provide an explicit approximation for the length of a torus knot.

Paper Structure

This paper contains 20 sections, 2 theorems, 42 equations, 11 figures.

Table of Contents

  1. Introduction and Goals
  2. Background on Torus Knots and Electric Fields
  3. Torus Knots
  4. The Electric Potential
  5. The Electric Field
  6. Branch cuts in Complex Analysis
  7. Numerical Analysis of Fields along the z-axis
  8. Fields in the x and y directions
  9. Field in the z-direction
  10. Critical points of the electric potential along the z-axis
  11. Numerical Calculations of Integrals
  12. Numerical Observations
  13. Analytic Calculations on the Length of the Torus Knot
  14. The Length Integral
  15. Finding Roots and Poles
  16. ...and 5 more sections

Key Result

Proposition 4.1

The electric field from a point along the $z$-axis is parallel to the $z$-axis.

Figures (11)

  • Figure 1: A trefoil knot from the above parametrization at various angles
  • Figure 2: A (3,8)-torus knot from the above parametrization at various angles
  • Figure 3: A (3,2)-torus knot from the above parametrization at various angles
  • Figure 4: A possible contour for $\sqrt z$
  • Figure 5: Electric field at $z$-coordinates from $-3$ to $3$ for the trefoil
  • ...and 6 more figures

Theorems & Definitions (8)

  • Definition 2.1
  • Definition 3.1
  • Definition 3.2: branch
  • Definition 3.3: branch
  • Definition 3.4: branch
  • Proposition 4.1
  • Theorem 4.2
  • proof