The Curvature of Lattice Knots

E. J. Janse van Rensburg; S. D. Promislow

doi:10.1142/s0218216599000328

The Curvature of Lattice Knots

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216599000328 ◽

1999 ◽

Vol 08 (04) ◽

pp. 463-490 ◽

Cited By ~ 15

Author(s):

E. J. Janse van Rensburg ◽

S. D. Promislow

Keyword(s):

Total Curvature ◽

Cubic Lattice ◽

Minimal Curvature

A result of Milnor [1] states that the infimum of the total curvature of a tame knot K is given by 2πμ(K), where μ(K) is the crookedness of the knot K. It is also known that μ(K)=b(K), where b(K) is the bridge index of K [2]. The situation appears to be quite different for knots realised as polygons in the cubic lattice. We study the total curvature of lattice knots by developing algebraic techniques to estimate minimal curvature in the cubic lattice. We perform simulations to estimte the minimal curvature of lattice knots, and conclude that the situation is very different than for tame knots in ℛ3.

Download Full-text

Total curvature, ropelength and crossing number of thick knots

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004107000151 ◽

2007 ◽

Vol 143 (1) ◽

pp. 41-55 ◽

Cited By ~ 6

Author(s):

Y. DIAO ◽

C. ERNST

Keyword(s):

Total Curvature ◽

Crossing Number ◽

Cubic Lattice ◽

The Relationship

AbstractWe first study the minimum total curvature of a knot when it is embedded on the cubic lattice. Let $\mathcal{K}$ be a knot or link with a lattice embedding of minimum total curvature $\tau(\mathcal{K})$ among all possible lattice embeddings of $\mathcal{K}$. We show that there exist positive constants c1 and c2 such that $c_1\sqrt{Cr(\mathcal{K})}\le \tau(\mathcal{K})\le c_2 Cr(\mathcal{K})$ for any knot type $\mathcal{K}$. Furthermore we show that the powers of $Cr(\mathcal{K})$ in the above inequalities are sharp hence cannot be improved in general. Our results and observations show that lattice embeddings with minimum total curvature are quite different from those with minimum or near minimum lattice embedding length. In addition, we discuss the relationship between minimal total curvature and minimal ropelength for a given knot type. At the end of the paper, we study the total curvatures of smooth thick knots and show that there are some essential differences between the total curvatures of smooth thick knots and lattice knots.

Download Full-text

A MINIMAL LINK ON THE CUBIC LATTICE

Journal of Knot Theory and Its Ramifications ◽

10.1142/s021821650200155x ◽

2002 ◽

Vol 11 (02) ◽

pp. 165-172 ◽

Cited By ~ 2

Author(s):

C. ERNST ◽

M. PHIPPS

Keyword(s):

Cubic Lattice ◽

Minimal Curvature

The main theorem in this article shows that the link [Formula: see text] cannot be realized with fewer than 28 steps on the cubical lattice. There are at least two very different 28-step realizations of [Formula: see text] on the cubical lattice in which the length of the components of the link differs. The two 28-step realizations of [Formula: see text] do not have minimal curvature. A realization with a minimal curvature of 6.5 π of the link [Formula: see text] with 30 steps is also shown.

Download Full-text