Youngsik Huh

Lattice stick number s L (K) is defined to be the minimal number of sticks required to construct a polygonal representation of the knot K in the cubic lattice. In this paper, we give lattice stick numbers of small knots such as 3 1 and 4 1 . More precisely we prove that s L (3 1 ) = 12 and s L (K) ≥ 14 for any other non-trivial knot K.

We investigate the number of knots and links in linear embeddings of <TEX>$K_6$</TEX>, the complete graph with 6 vertices. Concretely, we show that any linear embedding of <TEX>$K_6$</TEX> contains either only one Hopf link, or three Hopf links and one trefoil knot.

Monitoring technique for detection of early damage in the wind turbine blade has been investigated using PVDF film sensor and strain gages. The trailing edge component with shear web was prepared by sectioning the full-scale blade with a box spa. Several PVDF film sensors with dimension of 10x10x0.054 mm3 and strain gages were installed on shear web and skin of the component and the behaviour of these sensors were examined under opening and closing loads. From the test, optimal location for dete

Negami found an upper bound on the stick number s(K) ofa nontrivial knot K in terms of the minimal crossing number c(K) oftheknot, which is s(K) ≤ 2c(K). Furthermore, McCabe proved that s(K) ≤ c(K) +3 for a 2-bridge knot or link, except in the cases of the unlink and the Hopf link. In this paper we construct any 2-bridge knot or link K of at least six crossings by using only c(K) + 2 straight sticks. This gives a new upper bound on stick numbers of 2-bridge knots and links in terms of crossing n

We investigate the number of knots and links in linear embeddingsof K_6, the complete graph with 6 vertices. Concretely, we showthat any linear embedding of K_6 contains either only one Hopflink, or three Hopf links and one trefoil knot.

A generic map from a finite graph to the 2-space is called identifiable if any two embeddings of the graph into the 3-space obtained by lifting the map with respect to the natural projection from the 3-space to the 2-space are ambient isotopic in the 3-space. We show that only planar graphs have identifiable maps. We characterize the identifiable maps for some planar graphs.

A finite set of nontrivial θ n -curves is shown to be minimal among those which produce all projections of nontrivial θ n -curves.

A θ-curve is called almost trivial if it does not contain any non-trivial knot. A θ-curve is called strongly almost trivial if it has a planar projection which does not contain a projection of any non-trivial knot. In this paper, we introduce a method to present strongly almost trivial θ-curves. We also give an almost trivial θ-curve which may not be strongly almost trivial.

In 1983 Conway and Gordon proved that any embedding of the complete graph K 7 into ℝ 3 contains at least one nontrivial knot as its Hamiltonian cycle. After their work knots (also links) are considered as intrinsic properties of abstract graphs, and numerous subsequent works have been continued until recently. In this paper, we are interested in knotted Hamiltonian cycles in linear embedding of K 7 . Concretely it is shown that any linear embedding of K 7 contains at most three figure-8 knots.

Knots are commonly found in molecular chains such as DNA and proteins, and they have been considered to be useful models for structural analysis of these molecules. One interested quantity is the minimum number of monomers necessary to realize a molecular knot. The minimum lattice length Len(K) of a knot K indicates the minimum length necessary to construct K in the cubic lattice. Another important quantity in physical knot theory is the ropelength which is one of the knot energies measuring the

Abstract In this paper we investigate a ropelength-minimizing conformation of 4-strand superhelical strings whose axial curves constitute the standard double helix. In Huh et al (2016 J. Phys. A: Math. Theor . 49 415205) the authors found a specific conformation of standard double helix which was mathematically shown to be the unique ropelength-minimizing conformation. Adopting the conformation as axial curves we present a parametrization of superhelical curves so that the resulting shape is con

For a certain infinite family of knots or links, we study the growth power ratios of their stick number, lattice stick number, minimum lattice length and minimum ropelength compared with their minimum crossing number c(K) for every . It is known that the stick number and lattice stick number grow between the and linear power of the crossing number, and minimum lattice length and minimum ropelength grow with at least the power of crossing number (which is called the four-thirds power law). Furthe

The normalized Yamada polynomial, [Formula: see text], is a polynomial invariant in variable A for θ-curves. In this work, we show that the coefficients of [Formula: see text] which is obtained by replacing A with e x = ∑ x n /n! are finite-type invariants for θ-curves although the coefficients of original [Formula: see text] are not finite-type. A similar result can be obtained in the case of Yokota polynomial for θ-curves.