[Paper Review] Bounds on Übercrossing and Petal Numbers for Knots

Colin Adams, Orsola Capovilla‐Searle+6 more|arXiv (Cornell University)|Nov 3, 2013

Advanced Numerical Analysis Techniques3 citations

TL;DR

This paper establishes tight bounds on übercrossing and petal numbers for knots, proving that the übercrossing number of (r,r+1)-torus knots is exactly 2r and deriving relationships between these invariants and classical knot invariants like crossing number, bridge number, and unknotting number. It also shows that minimal petal projections can be obtained via the petal algorithm, but not necessarily from minimal crossing projections, especially for 2-braid knots.

ABSTRACT

An $n$-crossing is a point in the projection of a knot where $n$ strands cross so that each strand bisects the crossing. An übercrossing projection has a single $n$-crossing and a petal projection has a single $n$-crossing such that there are no loops nested within others. The übercrossing number, $ ext{ü}(K)$, is the smallest $n$ for which we can represent a knot $K$ with a single $n$-crossing. The petal number is the number of loops in the minimal petal projection. In this paper, we relate the übercrossing number and petal number to well-known invariants such as crossing number, bridge number, and unknotting number. We find that the bounds we have constructed are tight for $(r, r+1)$-torus knots. We also explore the behavior of übercrossing number under composition.

Motivation & Objective

• To establish tight bounds on the übercrossing number and petal number of knots using classical invariants such as bridge number, crossing number, and unknotting number.
• To determine the exact übercrossing number for the infinite family of (r,r+1)-torus knots, showing it equals 2r.
• To analyze the behavior of the übercrossing number under knot composition, providing an upper bound in terms of individual übercrossing and petal numbers.
• To investigate the relationship between petal number and unknotting number, proving a specific upper bound that is tight only for (r,r+1)-torus knots.
• To examine the effectiveness of the petal algorithm in generating minimal petal projections, showing it does not always succeed when applied to minimal crossing projections.

Proposed method

• Derive a lower bound on the übercrossing number using the bridge number, proving ü(K) ≥ 2b(K), and verify its tightness for (r,r+1)-torus knots.
• Use the petal algorithm to transform double-crossing projections into pre-petal and then petal projections, reversing the process to construct double-crossing projections that yield minimal petal projections.
• Apply the petal algorithm to minimal crossing projections of 2-braid knots and demonstrate that the resulting petal projections are not minimal, thereby showing that minimal crossing projections are not sufficient for minimal petal output.
• Establish an upper bound on the unknotting number: u(K) ≤ (p(K)−1)(p(K)−3)/8, and prove this bound is achieved only by (r,r+1)-torus knots.
• Relate petal number to traditional crossing number via the inequality c(K) ≤ (p²−2p−3)/4, and show equality holds for (r,r+1)-torus knots.
• Use geometric isotopy and axis-based projection techniques to analyze the number of times a knot crosses a central axis A in übercrossing projections, linking this to the number of local extrema and the übercrossing number.

Experimental results

Research questions

• RQ1 What is the exact übercrossing number for the (r,r+1)-torus knots, and is it tight with respect to the bridge number?
• RQ2 How does the übercrossing number behave under knot composition, and can we bound it using the petal and übercrossing numbers of the summands?
• RQ3 What is the relationship between the petal number and the unknotting number, and for which knots is the upper bound tight?
• RQ4 Can the petal algorithm always generate a minimal petal projection when applied to any double-crossing projection, and is a minimal crossing projection sufficient for this?
• RQ5 What is the minimal crossing number of a projection that generates a minimal petal projection for a 2-braid knot, and how does it compare to the standard minimal crossing projection?

Key findings

• The übercrossing number of the (r,r+1)-torus knot T_{r,r+1} is exactly 2r, and this value matches the lower bound of 2b(K), proving the bound is tight for this family.
• The unknotting number satisfies u(K) ≤ (p(K)−1)(p(K)−3)/8, and this bound is achieved precisely for (r,r+1)-torus knots, implying their minimal petal permutations are unique up to equivalence.
• The crossing number satisfies c(K) ≤ (p²−2p−3)/4, and equality holds for (r,r+1)-torus knots, showing the bound is tight.
• For any two knots K₁ and K₂, the übercrossing number of their connected sum satisfies ü(K₁#K₂) ≤ min{ü(K₁)+p(K₂)−1, ü(K₂)+p(K₁)−1}, providing a useful bound for composite knots.
• The petal algorithm does not generate a minimal petal projection when applied to the minimal crossing projection of a 2-braid knot with c > 3 crossings, demonstrating that minimal crossing projections are not always sufficient for minimal petal output.
• There exists a double-crossing projection (with 3/2(c−1) crossings on three strands) that, when the petal algorithm is applied, yields the minimal petal projection of a 2-braid knot, showing that non-minimal projections can be more effective for this purpose.