Kishino's knot is not detected by the fundamental group or the bracket polynomial; these invariants cannot differentiate between Kishino's knot and the unknot. However, we can show that Kishino's knot is not equivalent to unknot by applying either the 3-strand bracket polynomial or the surface bracket polynomial. In this paper, we construct two non-trivial virtual knot diagrams, K D and K m , t...

1 Preliminaries A virtual knot diagram is presented by a knot diagram having virtual crossings as well as real crossings in Fig. 1. Two virtual knot diagrams are equivalent if one can be obtained from the other by a finite sequence of generalized Reidemeister moves in Fig. 2. The equivalence class of virtual knot diagrams modulo the generalized Reidemeister moves is called a virtual knot. Virtu...

Molecular dynamics studies within a coarse-grained, structure-based model were used on two similar proteins belonging to the transcarbamylase family to probe the effects of the knot in the native structure of a protein. The first protein, N-acetylornithine transcarbamylase, contains no knot, whereas human ormithine transcarbamylase contains a trefoil knot located deep within the sequence. In ad...

Equivalence classes of virtual knot diagrams are in a one to one correspondence with knot diagrams (decorated immersions of S1) in orientable, closed surfaces modulo stable handle equivalence and Reidemeister moves. Each virtual knot diagram corresponds to a diagram in a unique minimal surface. If a virtual knot diagram is equivalent to a classical knot diagram then this minimal surface is a sp...

Abstract A holonomic knot is a knot in 3-space which arises as the 2-jet extension of a smooth function on the circle. A holonomic knot associated to a generic function is naturally framed by the blackboard framing of the knot diagram associated to the 1-jet extension of the function. There are two classical invariants of framed knot diagrams: the Whitney index (rotation number) W and the self ...

The derived group of a permutation representation, introduced by R.H. Crowell, unites many notions of knot theory. We survey Crowell’s construction, and offer new applications. The twisted Alexander group of a knot is defined. Using it, we obtain twisted Alexander modules and polynomials. Also, we extend a well-known theorem of Neuwirth and Stallings giving necessary and sufficient conditions f...

The aim of this study was to evaluate and compare the mechanical properties of five suture materials on three knot configurations when subjected to different physical conditions. Five 5-0 (silk, polyamide 6/66, polyglycolic acid, glycolide-e-caprolactone copolymer, polytetrafluoroethylene) suture materials were used. Ten samples per group of each material were used. Three knot configurations we...

We introduce the notions of “k-connected-slice” and “π1-slice”, interpolating between “homotopy ribbon” and “slice”. We show that every high-dimensional knot group π is the group of an (n− 1)-connected-slice n-knot for all n ≥ 3. However if π is the group of an n-connected-slice n-knot the augmentation ideal I(π) must have deficiency 1 as a module. If moreover n = 2 and π is finitely generated ...

BACKGROUND/AIM We report here an instrument designed by Dr Hasan Bdrekci that provides support to the knot to facilitate knot tying in laparoscopic procedures. We call the device the 'B6rekci knot-supporting instrument' and the technique 'B6rekci's knot technique' MATERIALS AND METHODS To evaluate the efficacy of this instrument, 17 surgeons performing laparoscopic surgery tied 3 knots using ...

A knot is an embedding of a circle S in a 3-manifold M , usually taken to be R or S. In the 1920’s Alexander and Briggs [2, §4] and Reidemeister [23] observed that questions about ambient isotopy of polygonal knots in R can be reduced to combinatorial questions about knot diagrams. These are labeled planar graphs with overcrossings and undercrossings marked, representing a projection of the kno...