Immersed flat ribbon knots

TL;DR

This work develops a variational framework for immersed planar flat ribbon knots by embedding the space of ribbon diagrams into a tractable disk-diagram space, enabling the analysis of length minimisers. It proves existence of length minimisers in disk space and shows that, when minimisers satisfy ribbon conditions, they solve the ribbonlength problem; it also identifies cases where disk-space minimisers fail to be ribbon. The authors compute minimal ribbonlengths for several diagrams (including the unknot, trefoil, Hopf link, and Whitehead-related families) and derive a bound on crossing numbers for diagrams achieving minimal ribbonlength, highlighting both the power and limitations of the disk-space approach. The results illuminate the relationship between disk-space minimisers and true ribbonlength minimisers, offering concrete criteria for verifying optimal diagrams across knot types and suggesting practical avenues for further refinement and applications in planar knot theory.

Abstract

We study the minimum ribbonlength for immersed planar ribbon knots and links. Our approach is to embed the space of such knots and links into a larger more tractable space of disk diagrams. When length minimisers in disk diagram space are ribbon, then these solve the ribbonlength problem. We also provide examples when minimisers in the space of disk diagrams are not ribbon and state some conjectures. We compute the minimal ribbonlength of some small knot and link diagrams and certain infinite families of link diagrams. Finally we present a bound for the number of crossings for a diagram yielding the minimum ribbonlength of a knot or link amongst all diagrams.

Figures (18)

• Figure 1: Examples of minimal ribbonlength knot and link diagrams. From left to right: the trefoil, Hopf link and an element of an infinite family of link diagrams whose ribbonlength can be computed.
• Figure 2: The broken lines are the ribbon loop $\gamma$ (core) associated with the open ribbon $\Gamma$, while the arrows show a branch of $\gamma$. The grey areas are neighbourhoods around self-intersections. From left to right: a ribbon self-intersecting at an isolated double point. A ribbon self-intersecting in a double interval. A non-crossing self-intersection at a double point. A non-crossing self-intersection in a double interval. Finally, a self-intersection not satisfying the crossing condition as the ribbon self-intersects but there are no associated double points or intervals of the core.
• Figure 3: For a double point $p$ of $\gamma$ there is a neighbourhood $U_p$ so that $\Gamma^{-1}(U_p)=D_1\cup D_2$, where $D_1, D_2$ are disjoint open disks which separate the annulus into contractible components. Note that $\Gamma(s,0)=\gamma(s)$, for $s\in S^1$.
• Figure 4: (a) the limit of a sequence of double points for the $\gamma_i$ can lie in a double interval of $\hat{\gamma}$. (b) two "close" double points $x,x´$ for $\hat{\gamma}$. (c) the tangents to $\lambda_1, \lambda_2$ may agree at $x'$. (d) a switch between the arcs $\lambda_1$ and $\lambda_2$.
• Figure 5: Top: Smoothing the corners at $x$. Bottom: An example where the centres of the unit circles are where the boundary of the ribbon $\hat{\Gamma}$ crosses itself.

Theorems & Definitions (28)

• Definition 2.1
• Definition 2.2
• Proposition 2.3
• proof
• Definition 2.4
• Definition 2.5
• Proposition 2.7
• proof
• Definition 2.9
• Theorem 2.10