Hamiltonian knot projections and lengths of thick knots

Abstract

For a knot or link K, L(K) denotes the rope length of K and Cr(K) denotes the crossing number of K. An important problem in geometric knot theory concerns the bound on L(K) in terms of Cr(K). It is well known that there exist positive constants c 1 , c 2 such that for any knot or link K, c 1 • (Cr(K)) 3/4 ≤ L(K) ≤ c 2 • (Cr(K)) 2 . In this paper, we prove that there exists a constant c > 0 such that for any knot or link K, L(K) ≤ c • (Cr(K)) 3/2 . This is done through the study of regular projections of knots and links as 4-regular plane graphs. We show that for any knot (or link) K there exists a knot (or link) K and a regular projection G of K such that K is of the same knot type as K, G has at most 4 • Cr(K) crossings, and G is a Hamiltonian graph. We then use this result to develop an embedding algorithm. Using this algorithm, we are able to embed any knot or link K into the simple cubic lattice such that the length of the embedded knot is of order at most O((Cr(K)) 3/2 ). This result in turn establishes the above mentioned upper bound on L(K) for smooth knots and links. Regular projection graphsIn this section, we introduce some basic concepts and results in knot theory and graph theory. In addition, some new terms are defined as they will be needed in our discussions.A graph G consists of a set V (G), a set E(G), and an incidence relation which says that every element of E(G) is incident with two elements of V (G). The elements of V (G) are called the vertices of G, and V (G) is called the vertex set of G. The elements of E(G) are called the edges of G, and E(G) is called the edge set of G.Let G be a graph. The degree of a vertex v of G is the number of edges of G incident with v. The graph G is called k-regular if every vertex of G is of degree k. Let e be an edge of G incident with vertices u and v. We say that e connects u and v, u and v are the ends of e, and u and v are adjacent. We also use uv or vu to denote e when there is only one edge connecting u and v. If u = v then e is called a loop edge.A graph H is a subgraph of a graph G if V (H) ⊂ V (G) and E(H) ⊂ E(G). In the case that V (H) = V (G), H is called a spanning subgraph of G. Two graphs G and H are said to be isomorphic if there exist bijections f :A path P of length k − 1, where k ≥ 2, is a graph which is isomorphic to the graph with vertex set {v 1 , v 2 , ..., v k } and edge set {v i v i+1 : i = 1, . . . , k − 1}. We say that v 1 and v k are the ends of P , P is from v 1 to v k , and P is between v 1 and v k . A cycle C of length k, where k ≥ 3, is a graph which is isomorphic to the graph with vertex set {v 1 , v 2 , ..., v k } and edge set {v i v i+1 : i = 1, . . . , k − 1} ∪ {v k v 1 }. A Hamilton cycle in a graph G is a spanning subgraph that is a cycle. A graph with a Hamilton cycle is said to be Hamiltonian.A geometric realization of a graph G in R 2 or R 3 is such that the vertices of G are represented by distinct points in the space, every edge of G is represented by a simple curve in R 2 or R 3 connecting the two points representing...

Total curvature, ropelength and crossing number of thick knots

“…It follows that K can be realized by a smooth knot of unit thickness with length at most b 1 ·(Cr(K)) 3/2 whose total curvature is bounded above by b 2 ·Cr(K) for some constant b 2 > 0. For details of the embedding construction we refer to [15]. This proves (i).…”

“…The proof of (i) relies on the lattice embedding of K on the cubic lattice as described in [15] and Theorem 2·5. It is proven in [15] that there exists a constant b 1 > 0 such that any knot type K can be embedded in the cubic lattice with a length at most b 1 (Cr(K)) 3/2 . Furthermore, the total curvature of such an embedding is of the order O(Cr(K)).…”

“…Furthermore, the total curvature of such an embedding is of the order O(Cr(K)). The construction in Theorem 2·5 uses a simplified version of the embedding given in [15], which would not achieve a ropelength bound of O(Cr(K) 3/2 ). Although the Hamiltonian cycle is embedded in a more complicated way in the construction used in [15], it still has an embedding length of O(Cr(K)).…”

On the normal form of knots

“…(This aspect of the problem has been addressed in many papers (see e.g. [3,4], and the references there); the solutions proposed in those papers appear under the denominations "ideal knot," "tight knot," "fat knot," "knot of minimal rope length," and so on.) In the present paper, we discuss the aesthetic merits and shortcomings of the normal forms obtained.…”

Ropelengths of closed braids

“…The case of upper bounds turns out to be harder and no sharp upper bounds have been found in general at the writing of this paper. The best upper bound known for the ropelength of any knot K is of the order O((Cr(K)) 3/2 ) [11]. However, it is not known whether this bound is sharp (it is generally believed that it is not).…”