[Paper Review] Linear Upper Bound on the Ribbonlength of Torus Knots and Twist Knots

Grace Tian|arXiv (Cornell University)|Sep 6, 2018

Geometric and Algebraic Topology10 references3 citations

TL;DR

This paper establishes a linear upper bound on the ribbonlength for all non-trivial torus knots and twist knots using grid diagrams to construct flat knotted ribbons. By thickening grid paths and optimizing folding strategies, the author improves upon prior quadratic bounds, proving that ribbonlength grows linearly with crossing number—supporting Kusner's conjecture and extending results to broader knot families via a novel grid-based method.

ABSTRACT

Knotted ribbons form an important topic in knot theory. They have applications in natural sciences, such as cyclic duplex DNA modeling. A flat knotted ribbon can be obtained by gently pulling a knotted ribbon tight so that it becomes flat and folded. An important problem in knot theory is to study the minimal ratio of length to width of a flat knotted ribbon. This minimal ratio is called the ribbonlength of the knot. It has been conjectured that the ribbonlength has an upper bound and a lower bound which are both linear in the crossing number of the knot. In the first part of the paper, we use grid diagrams to construct flat knotted ribbons and prove an explicit quadratic upper bound on the ribbonlength for all non-trivial knots. We then improve the quadratic upper bound to a linear upper bound for all non-trivial torus knots and twist knots. Our approach of using grid diagrams to study flat knotted ribbons is novel and can likely be used to obtain a linear upper bound for more general families of knots. In the second part of the paper, we obtain a sharper linear upper bound on the ribbonlength for nontrivial twist knots by constructing a flat knotted ribbon via folding the ribbon over itself multiple times to shorten the length.

Motivation & Objective

• To establish a linear upper bound on ribbonlength for non-trivial torus knots and twist knots, addressing a key open problem in knot theory.
• To improve upon previous quadratic upper bounds for ribbonlength by introducing a novel grid diagram-based construction method.
• To support Kusner's conjecture that ribbonlength is linearly bounded in the crossing number for all knots.
• To extend the applicability of grid diagrams to flat knotted ribbon constructions for broader families of knots.
• To develop a sharper linear bound for twist knots through multi-folded ribbon constructions.

Proposed method

• Constructing flat knotted ribbons by thickening horizontal and vertical paths in a grid diagram representation of the knot.
• Using the known relationship between grid index and crossing number to derive a quadratic upper bound on ribbonlength for all non-trivial knots.
• Applying geometric optimization to reduce ribbon length by folding the ribbon over itself multiple times, especially for twist knots.
• Employing Reidemeister moves to verify topological equivalence of the constructed ribbon to the original knot.
• Deriving explicit formulas for ribbon length in terms of width and crossing parameters, including trigonometric and algebraic expressions involving the golden ratio.
• Using the grid index $ g(K) \leq c(K) + 2 $ to explore potential linear bounds in terms of crossing number $ c(K) $.

Experimental results

Research questions

• RQ1 Can a linear upper bound on ribbonlength be established for all non-trivial torus knots?
• RQ2 Can the quadratic upper bound for ribbonlength on $(2,q)$ torus knots be improved to a linear bound?
• RQ3 Can a sharper linear upper bound be achieved for twist knots through optimized ribbon folding?
• RQ4 Is the grid diagram approach generalizable to other families of knots beyond torus and twist knots?
• RQ5 Does the ribbonlength of a knot scale linearly with its crossing number, as conjectured by Kusner?

Key findings

• A linear upper bound on ribbonlength is proven for all non-trivial torus knots, improving upon prior quadratic bounds for $(2,q)$ torus knots.
• For twist knots, a sharper linear upper bound is achieved by folding the ribbon over itself multiple times, reducing length more effectively than grid-based thickening alone.
• The ribbonlength of twist knots $J(2,n)$ is bounded above by $ \left[\frac{\sqrt{5}+1}{2}n + \frac{9+\sqrt{5}}{2} + \sqrt{\frac{5+\sqrt{5}}{2}} \right]w $, demonstrating explicit linear scaling.
• The grid diagram method provides a novel and generalizable framework for constructing flat knotted ribbons, enabling linear bounds across families of knots.
• The results support Kusner’s conjecture that ribbonlength is linearly bounded in the crossing number for all knots.
• The method extends to mirror images and related knots, as shown by the equivalence of $J(2,-(n-1))$ to the mirror of $J(2,n)$.