Total curvature, ropelength and crossing number of thick knots

Abstract

We 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.

Links with small lattice stick numbers

“…To find the exact values of the lattice stick number of these knots, they used a lower bound on lattice stick number in terms of bridge number, s L (K) 6b(K), which was proved in [10]. Furthermore Diao and Ernst [3] found a general lower bound in terms of the crossing number c(K), which is s L (K) 3 √ c(K) + 1 + 3 for a non-trivial knot K. Recently Hong et al [5] found a general upper bound s L (K) 3c(K) + 2, and moreover s L (K) 3c(K) − 4 for a non-alternating prime knot.…”

Links with small lattice stick numbers

“…Each component contains at least two x-sticks, two y-sticks and two z-sticks since both are nonplanar. We may say that (L) is one of (4, 4, 4), (4,4,5), (4,4,6) or (4,5,5). Now consider the projection of L onto an xy-plane.…”

“…To find the exact values of the lattice stick number of these knots, they used a lower bound on lattice stick number in terms of bridge number, s L (K) ≥ 6b(K), which was proved in [11]. Furthermore Diao and Ernst [4] found a general lower bound in terms of crossing number c(K) which is s L (K) ≥ 3 c(K) + 1 + 3 for a nontrivial knot K. Recently Hong, No and Oh [6] found a general upper bound s L (K) ≤ 3c(K)+2, and moreover s L (K) ≤ 3c(K) − 4 for a non-alternating prime knot.…”

Circuit presentation and lattice stick number with exactly four z-sticks

“…To find the exact values of the lattice stick number of these knots, they used a lower bound on lattice stick number in terms of bridge number, s L (K) ≥ 6b(K), which was proved by Janse van Rensburg and Promislow [10]. Furthermore Diao and Ernst [4] found a general lower bound in terms of crossing number c(K) which is s L (K) ≥ 3 c(K) + 1+3 for a nontrivial knot K. Hong, No and Oh [5] found a general upper bound s L (K) ≤ 3c(K)+2, and moreover s L (K) ≤ 3c(K)−4 for a non-alternating prime knot. They [6] also showed that s L (2 2 1 ) = 8, s L (2 2 1 ♯2 2 1 ) = s L (6 3 2 ) = s L (6 3 3 ) = 12, s L (4 2 1 ) = 13, s L (5 2 1 ) = 14 and any other non-split links have stick numbers at least 15.…”