[Paper Review] A Method for Unknotting Torus Knots

Vikash Siwach, Madeti Prabhakar|arXiv (Cornell University)|Jul 20, 2012

Geometric and Algebraic Topology3 references3 citations

TL;DR

This paper presents a systematic method to identify minimal unknotting crossing changes in torus knots and links using braid group theory and Markov moves. It provides a constructive algorithm to locate exactly (p−1)(q−1)/2 crossings in the braid closure of B(p,q) whose sign flips yield the unknot, and proves the symmetry K(p,q) ∼ K(q,p) via Markov equivalence, offering a concrete, patterned alternative to brute-force crossing selection.

ABSTRACT

Unknotting numbers for torus knots and links are well known. In this paper, we present a method for determining the position of unknotting number crossing changes in a toric braid B(p, q) such that the closure of the resultant braid is equivalent to the trivial knot or link. Also, we provide a simple proof for the important result K(p, q) \sim K(q, p) using the results obtained from this method.

Motivation & Objective

• To develop a constructive method for identifying the precise positions of (p−1)(q−1)/2 crossing changes in the braid B(p,q) that unknot the closure of the braid.
• To provide a simple, pattern-based proof of the symmetry K(p,q) ∼ K(q,p) for torus knots using the proposed unknotting method.
• To extend the minimal unknotting crossing data method from torus knots to torus links, particularly when gcd(p,q) = d > 1.
• To offer a computationally efficient alternative to generic unknotting procedures that rely on ascending/descending diagram conversion without pattern.

Proposed method

• The method uses the braid group B_p and the standard toric braid B(p,q) = (σ₁σ₂⋯σ_{p−1})^q, whose closure is the torus knot K(p,q).
• It defines a specific set of crossings, called the U-crossing data, based on a recursive structure involving σ_i and their inverses, which are candidates for unknotting.
• The method applies Markov moves to transform the braid after crossing changes, showing that the resulting braid is Markov equivalent to a simpler braid, ultimately proving it is trivial.
• For torus links with gcd(p,q) = d > 1, the method uses Euclid’s algorithm to decompose the braid into simpler components, each of which is handled via the same unknotting logic.
• The algorithm is implemented in MATLAB (Appendix A), taking p and q as input to output two distinct minimal unknotting crossing sets.
• The proof of symmetry K(p,q) ∼ K(q,p) relies on showing that changing crossings in B(p,a) based on U(B(p,a)) yields a braid Markov equivalent to B(a,p−a).

Experimental results

Research questions

• RQ1 How can one systematically identify the exact positions of (p−1)(q−1)/2 crossings in the braid B(p,q) whose sign flips result in a trivial knot?
• RQ2 Can the symmetry K(p,q) ∼ K(q,p) for torus knots be proven using braid group relations and Markov moves rather than topological invariants?
• RQ3 What is the minimal unknotting crossing data for torus links K(p,q) when gcd(p,q) > 1, and how can it be constructed from simpler components?
• RQ4 Is there a patterned, non-brute-force method to find minimal unknotting crossing data in torus knots, avoiding the inefficiency of ascending/descending diagram conversion?
• RQ5 How can the unknotting process for torus knots be extended to torus links using recursive decomposition and Markov equivalence?

Key findings

• The U-crossing data for B(p,q) is a minimal unknotting crossing data if and only if q ≡ 1 or p−1 (mod p), providing a precise condition for validity.
• The unknotting number of a torus knot K(p,q) is exactly (p−1)(q−1)/2, and this number of crossing changes suffices to unknot the closure of B(p,q) when applied at the U-crossing data positions.
• The method proves that K(p,q) ∼ K(q,p) via Markov equivalence: changing crossings in B(p,a) based on U(B(p,a)) yields a braid Markov equivalent to B(a,p−a).
• For torus links K(p,p), the U-crossing data of B(p,p) is minimal and yields a trivial link, with |U(B(p,p))| = (p−1)p/2 crossings.
• For general torus links K(p,q) with gcd(p,q) = d > 1, the method decomposes the braid using Euclid’s algorithm and constructs minimal unknotting data by combining results from subcomponents.
• The MATLAB implementation in Appendix A successfully computes two distinct minimal unknotting crossing sets for any K(p,q), confirming the theoretical results computationally.