In this paper, we introduce a bisected vertex leveling of a plane graph. Using this planar embedding, we present elementary proofs of the well-known upper bounds in terms of the minimal crossing number on braid index [Formula: see text] and arc index [Formula: see text] for any knot or non-split link [Formula: see text], which are [Formula: see text] and [Formula: see text]. We also find a quadratic upper bound of the minimal crossing number of delta diagrams of [Formula: see text].
Knots and embedded graphs are useful models for simulating polymer chains. In particular, a theta curve motif is present in a circular protein with internal bridges. A theta-curve is a graph embedded in three-dimensional space which consists of three edges with shared endpoints at two vertices. If we cannot continuously transform a theta-curve into a plane without intersecting its strand during the deformation, then it is said to be nontrivial. A Brunnian theta-curve is a nontrivial theta-curve that becomes a trivial knot if any one edge is removed. In this paper we obtain qualitative results of these theta-curves, using the lattice stick number which is the minimal number of sticks glued end-to-end that are necessary to construct the theta-curve type in the cubic lattice. We present lower bounds of the lattice stick number for nontrivial theta-curves by 14, and Brunnian theta-curves by 15.
The lattice stick number of knots is defined to be the minimal number of straight sticks in the cubic lattice required to construct a lattice stick presentation of the knot. We similarly define the lattice stick number sL(G) of spatial graphs G with vertices of degree at most six (necessary for embedding into the cubic lattice), and present an upper bound in terms of the crossing number c(G)where G has e edges, v vertices, s cut-components, b bouquet cutcomponents, and k knot components.
The ribbonlength Rib[Formula: see text] of a knot [Formula: see text] is the infimum of the ratio of the length of any flat knotted ribbon with core [Formula: see text] to its width. A twisted torus knot [Formula: see text] is obtained from the torus knot [Formula: see text] by twisting [Formula: see text] adjacent strands [Formula: see text] full twists. In this paper, we show that the ribbonlength of [Formula: see text] is less then or equal to [Formula: see text] where [Formula: see text] and [Formula: see text] are positive. Furthermore, if [Formula: see text], then the ribbonlength of [Formula: see text] is less then or equal to [Formula: see text].
A ribbon is a two-dimensional object with one-dimensional properties which is related with geometry, robotics and molecular biology. A folded ribbon structure provides a complex structure through a series of folds. We focus on a folded ribbon with knotted core. The folded ribbonlength Rib(K) of a knot K is the infimum of the quotient of length by width among the ribbons representing a knot type of K. This quantity tells how efficiently the folded ribbon is realized. Kusner conjectured that folded ribbonlength is bounded by a linear function of the minimal crossing number c(K). In this paper, we confirm that the folded ribbonlength of a 2-bridge knot K is bounded above by 2c(K)+2. 2020 Mathematics Subject Classification. 57K10. Key words and phrases. folded ribbonlength, folded ribbon knot, 2-bridge knot.
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