TL;DR
This paper generalizes the straight and contained straight numbers for all knots, proves their well-definedness, explores their relations to other knot invariants, and computes these numbers for knots with up to 10 crossings.
Contribution
It extends the definitions of straight and contained straight numbers to all knots, proves their properties, and provides comprehensive computations for knots with up to 10 crossings.
Findings
Straight numbers are well-defined for all knots.
Relations between straight numbers and crossing/petal numbers are established.
Complete tables of straight numbers for knots with up to 10 crossings are provided.
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.
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Taxonomy
TopicsGeometric and Algebraic Topology · Connective tissue disorders research