Stick index of knots and links in the cubic lattice
Colin Adams, Michelle Chu, Thomas Crawford, Stephanie Jensen Kyler, Siegel, Liyang Zhang
TL;DR
This paper investigates the minimal number of straight segments needed to construct knots and links in a cubic lattice, providing exact values for certain classes and establishing bounds relating to other invariants.
Contribution
It introduces the cubic lattice stick index for knots and links, computes it for various classes, and explores how composition and satellites affect this invariant.
Findings
Cubic lattice stick index computed for all (p,p+1)-torus knots.
Methods to derive the index for large classes of knots using composition and satellites.
Bounds relating cubic lattice stick index to other knot invariants.
Abstract
The cubic lattice stick index of a knot type is the least number of sticks necessary to construct the knot type in the 3-dimensional cubic lattice. We present the cubic lattice stick index of various knots and links, including all (p,p+1)-torus knots, and show how composing and taking satellites can be used to obtain the cubic lattice stick index for a relatively large infinite class of knots. Additionally, we present several bounds relating cubic lattice stick index to other known invariants.
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