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Ribbonlength upper bounds for small crossing knots and links

Zhicheng Chen, Elizabeth Denne, Kyle Patterson, Timi Patterson

TL;DR

This work advances the folded ribbonlength program by introducing a wrap-based construction that concentrates half-twists into compact square regions, enabling explicit, efficient realizations of several knot and link families. It proves new upper bounds for the infimal folded ribbonlength in three key families: $\operatorname{Rib}([T(2,q)]) \le q+3$ for $(2,q)$-torus links, $\operatorname{Rib}((T_n)_w) = n+6$ for twist knots, and $\operatorname{Rib}([P(p,q,r)]) \le |p|+|q|+|r|+6$ for pretzel links, with corollaries including a bound of $8$ for the figure-8 knot. The constructions yield tight or near-tight bounds for small-crossing knots and links, improving on universal bounds and offering concrete, constructive realizations. Together, these results sharpen our understanding of how ribbonlength scales with crossing structure and support the broader ribbonlength-crossing number program.

Abstract

Given a thin strip of paper, tie a knot, connect the ends, and flatten into the plane. This is a physical model of a folded ribbon knot in the plane, first introduced by Louis Kauffman. We study the folded ribbonlength of these folded ribbon knots, which is defined as the knot's length-to-width ratio. The {\em ribbonlength problem} asks to find the infimal folded ribbonlength of a knot or link type. By finding new methods of creating folded ribbon knots, we improve upon existing upper bounds for the folded ribbonlength of $(2,q)$-torus links, twist knots, and pretzel links. These give the best known bounds to date for small crossing knots in these families. For example, there is a folded ribbonlength twist knot $T_n$ with folded ribbonlength $\text{Rib}(T_n) = n +6$. Applying this to the figure-eight knot $T_2$ yields a folded ribbonlength $\text{Rib}(T_2)= 8$, which we conjecture is the infimum.

Ribbonlength upper bounds for small crossing knots and links

TL;DR

This work advances the folded ribbonlength program by introducing a wrap-based construction that concentrates half-twists into compact square regions, enabling explicit, efficient realizations of several knot and link families. It proves new upper bounds for the infimal folded ribbonlength in three key families: for -torus links, for twist knots, and for pretzel links, with corollaries including a bound of for the figure-8 knot. The constructions yield tight or near-tight bounds for small-crossing knots and links, improving on universal bounds and offering concrete, constructive realizations. Together, these results sharpen our understanding of how ribbonlength scales with crossing structure and support the broader ribbonlength-crossing number program.

Abstract

Given a thin strip of paper, tie a knot, connect the ends, and flatten into the plane. This is a physical model of a folded ribbon knot in the plane, first introduced by Louis Kauffman. We study the folded ribbonlength of these folded ribbon knots, which is defined as the knot's length-to-width ratio. The {\em ribbonlength problem} asks to find the infimal folded ribbonlength of a knot or link type. By finding new methods of creating folded ribbon knots, we improve upon existing upper bounds for the folded ribbonlength of -torus links, twist knots, and pretzel links. These give the best known bounds to date for small crossing knots in these families. For example, there is a folded ribbonlength twist knot with folded ribbonlength . Applying this to the figure-eight knot yields a folded ribbonlength , which we conjecture is the infimum.
Paper Structure (7 sections, 8 theorems, 7 equations, 14 figures)

This paper contains 7 sections, 8 theorems, 7 equations, 14 figures.

Table of Contents

  1. Introduction to Folded Ribbon Knots and Links
  2. Half-twists and link families
  3. The wrap method applied to $(2,q)$ torus links
  4. Ribbonlength of $(2,q)$-torus links
  5. The wrap method applied to twist knots
  6. Ribbonlength of pretzel links
  7. Acknowledgments

Key Result

Lemma 3

In the wrap method Construction const:wrap, we see $n$ half-twists can be constructed using at least $|n|+2$ units of ribbonlength and the corresponding knot diagram has $|n|+3$ sticks.

Figures (14)

  • Figure 1: Starting with a polygonal trefoil knot diagram (left), we create a folded ribbon trefoil knot (center). On the right, the pentagon shape arises when this particular folded ribbon trefoil knot is "pulled tight".
  • Figure 2: A $+3$ twist box on the left, a $-2$ twist-box in the center, and a $(2,q)$-torus link on the right.
  • Figure 3: A $T_5$ twist knot on the left, and an arbitrary $T_n$ twist knot on the right.
  • Figure 4: On the left, an arbitrary $P(p,q,r)$ pretzel link, and on the right a $P(3, 1, -2)$ pretzel knot.
  • Figure 5: Two different viewpoints of half-twists.
  • ...and 9 more figures

Theorems & Definitions (18)

  • Remark 1
  • Lemma 3
  • proof
  • Theorem 6
  • proof
  • Conjecture 7
  • Conjecture 8
  • Theorem 10
  • proof
  • Corollary 11
  • ...and 8 more