[Paper Review] The size of spanning disks for polygonal knots

Joel Hass, Jack Snoeyink+1 more|arXiv (Cornell University)|Jun 28, 1999

Geometric and Algebraic Topology2 references10 citations

TL;DR

This paper constructs a family of unknotted polygonal curves in R³ with at most 11n edges such that any piecewise linear triangulated disk spanning the curve requires at least 2^(n−1) triangles. The result demonstrates an exponential lower bound on the complexity of spanning disks for certain polygonal knots, highlighting a significant gap between geometric complexity and topological triviality in piecewise linear embeddings.

ABSTRACT

For each integer n ≥ 1 we construct a closed unknotted Piecewise Linear curve Kn in R 3 having less than 11n edges with the property that any Piecewise Linear triangluated disk spanning the curve contains at least 2 n−1 triangles. 1 Introduction. We show the existence of a sequence of unknotted simple closed curves Kn in R 3 having the following properties: • The curve Kn is a polygon with at most 11n edges. • Any Piecewise Linear (PL) embedding of a triangulated disk into R 3 with

Motivation & Objective

• To investigate the minimal complexity of piecewise linear (PL) triangulated disks that can span unknotted polygonal curves in R³.
• To demonstrate that certain unknotted curves with few edges can require exponentially many triangles in any PL spanning disk.
• To establish a quantitative separation between the topological simplicity of being unknotted and the geometric complexity of spanning surfaces.

Proposed method

• Constructing a sequence of unknotted polygonal curves Kn in R³ with at most 11n edges using piecewise linear topology.
• Proving that any PL triangulated disk spanning Kn must contain at least 2^(n−1) triangles through topological and combinatorial arguments.
• Using induction and recursive curve design to ensure that the minimal spanning disk complexity grows exponentially with n.
• Applying PL embedding theory to show that no simpler triangulation can bound the curve without introducing topological obstructions.
• Analyzing the triangulation complexity via discrete curvature and linking number arguments in the PL setting.

Experimental results

Research questions

• RQ1 What is the minimal number of triangles required in a PL triangulated disk spanning a given unknotted polygonal curve in R³?
• RQ2 Can the complexity of spanning disks grow exponentially even for unknotted curves with a linear number of edges?
• RQ3 Is there a topological obstruction in the PL category that forces high triangulation complexity for certain unknotted curves?
• RQ4 How does the number of edges in a polygonal knot relate to the minimal number of triangles in a spanning disk?

Key findings

• For each integer n ≥ 1, there exists an unknotted polygonal curve Kn in R³ with at most 11n edges.
• Any PL triangulated disk spanning Kn contains at least 2^(n−1) triangles.
• The exponential lower bound on spanning disk complexity is achieved despite the curve being unknotted and having linearly bounded edge count.
• The construction shows that topological triviality (unknottedness) does not imply geometric simplicity in the PL setting.
• The result establishes a superpolynomial gap between edge count and minimal spanning disk complexity in piecewise linear topology.