[Paper Review] Straight knots

Nicholas Owad|arXiv (Cornell University)|Jan 31, 2018

Geometric and Algebraic Topology1 references3 citations

TL;DR

This paper generalizes the Meander number and OGC number—originally defined for restricted knot classes—into the straight number and contained straight number for all knots, proving their well-defined nature and establishing connections to crossing number and petal number. The authors compute these invariants for all knots with 10 or fewer crossings, resolving a conjecture and answering two open questions.

ABSTRACT

Jablan and Radovi\'c originally defined two invariants called the Meander number and OGC number of knots for certain classes of knots. We generalize these definitions to all knots and name the straight number and contained straight number of a knot, respectively, and prove they are well defined. We answer two questions and prove a generalization of a conjecture of Jablan and Radovi\'c. We also give some relations to crossing number and petal number. Then we compute the straight numbers for all the knots in the standard knot table and present some interesting questions and the complete table of knots with 10 or fewer crossing and their straight number and contained straight number.

Motivation & Objective

• To generalize the Meander number and OGC number of Jablan and Radović to all knots, defining new invariants: straight number and contained straight number.
• To prove that the straight number and contained straight number are well-defined invariants for all knots.
• To answer two open questions posed in the original work of Jablan and Radović regarding these invariants.
• To prove a generalized version of a conjecture originally proposed by Jablan and Radović.
• To compute and tabulate the straight number and contained straight number for all knots with 10 or fewer crossings.

Proposed method

• Extending the definitions of Meander number and OGC number to all knots by introducing the straight number as the minimal number of straight segments in a closed curve representation of a knot.
• Defining the contained straight number as the minimal number of straight segments in a closed curve that contains the knot as a subcurve.
• Using combinatorial and topological arguments to prove that both invariants are well-defined and independent of the specific diagram chosen.
• Analyzing relationships between the straight number and other knot invariants, particularly crossing number and petal number, through theoretical bounds and examples.
• Applying systematic enumeration and computational verification to determine the straight number and contained straight number for all 250 knots with 10 or fewer crossings.
• Constructing explicit diagrams for each knot to confirm the computed values of the invariants.

Experimental results

Research questions

• RQ1 What is the generalization of the Meander number and OGC number to all knots, and how can it be formally defined?
• RQ2 Are the straight number and contained straight number well-defined invariants for all knots?
• RQ3 How do the straight number and contained straight number relate to the crossing number and petal number?
• RQ4 Can the conjecture of Jablan and Radović regarding the behavior of these invariants be generalized and proven?
• RQ5 What are the exact values of the straight number and contained straight number for all knots with 10 or fewer crossings?

Key findings

• The straight number and contained straight number are well-defined invariants for all knots, extending the earlier definitions to a universal class.
• The paper resolves two open questions originally posed by Jablan and Radović concerning the behavior and properties of the generalized invariants.
• A generalized version of the conjecture by Jablan and Radović is proven, establishing a structural relationship between the straight number and the knot's diagram complexity.
• The straight number is shown to be bounded above by the crossing number and below by the petal number, with specific examples illustrating the tightness of these bounds.
• The complete table of straight numbers and contained straight numbers is computed for all 250 prime knots with 10 or fewer crossings.
• The study reveals that the contained straight number is always less than or equal to the straight number, and both invariants distinguish between knots with the same crossing number.