相关论文: Slope lengths and generalized augmented links
We identify the space of tangentially straightened long knots in R^m, for m greater than or equal to 4, as the double loops on the space of derived operad maps from the associative operad into a version of the little m-disk operad. This…
We study certain linear representations of the knot group that induce augmentations of knot contact homology. This perspective on augmentations enhances our understanding of the relationship between the augmentation polynomial and the…
The Morse-Novikov number MN(L) of an oriented link L in the 3-sphere is the minimum number of critical points of a Morse map from the complement of L in the 3-sphere to the circle representing the class of a Seifert surface for L (e.g., the…
We study near-alternating links whose diagrams satisfy conditions generalized from the notion of semi-adequate links. We extend many of the results known for adequate knots relating their colored Jones polynomials to the topology of…
We show that, for an alternating knot, the ratio of the diameter of the set of boundary slopes to the crossing number can be arbitrarily large.
Among the knots that are the connected sum of two torus knots with cobordism distance 1, we characterize those that have 4-dimensional clasp number at least 2, and we show that their n-fold connected self-sum has 4-dimensional clasp number…
Length spectral rigidity is the question of under what circumstances the geometry of a surface can be determined, up to isotopy, by knowing only the lengths of its closed geodesics. It is known that this can be done for negatively curved…
We present computer simulations to examine probability distributions of gyration radius for the no-thickness closed polymers of N straight segments of equal length. We are particularly interested in the conditional distributions when the…
The ropelength of a knot is the quotient of its length by its thickness. We consider a family of energy functions for knots, depending on a power p, which approach ropelength as p increases. We describe a numerically computed trefoil knot…
Classical fully augmented links have explicit hyperbolic geometry, and have diagrams on the 2-sphere in the 3-sphere. We generalise to construct fully augmented links projected to the reflection surface of any 3-manifold obtained by…
This article gives a foundational account of various characterizations of framed links in the $3$-sphere.
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 $s_{L}(G)$ of spatial…
Let $\delta_0(P,k)$ denote the degree $k$ dilation of a point set $P$ in the domain of plane geometric spanners. If $\Lambda$ is the infinite square lattice, it is shown that $1+\sqrt{2} \leq \delta_0(\Lambda,3) \leq (3+2\sqrt2) \, 5^{-1/2}…
The amount and type of self-entanglement of DNA filaments is significantly affected by spatial confinement, which is ubiquitous in biological systems. Motivated by recent advancements in single DNA molecule experiments based on nanofluidic…
In this paper we prove that if a knot or link has a sufficiently complicated plat projection, then that plat projection is unique. More precisely, if a knot or link has a $2m$-plat projection, where $m$ is at least four, and height at least…
We give an upper bound on the distance between a degeneracy slope for a very full essential lamination and a boundary slope of an essential surface embedded in a compact, orientable, irreducible, atoroidal 3-manifold with incompressible…
We perform theoretical studies of stretching of 20 proteins with knots within a coarse grained model. The knot's ends are found to jump to well defined sequential locations that are associated with sharp turns whereas in homopolymers they…
We propose a new method for numerical calculation of link plynomials for knots given in 3 dimensions. We calculate derivatives of the Jones polynomial in a computational time proportional to $N^{\alpha}$ with respect to the system size $N$…
We introduce a new algebraic topological technique to detect non-fibred knots in the three sphere using the twisted Alexander invariants. As an application, we show that for any Seifert matrix of a knot with a nontrivial Alexander…
J. Davis showed that the topological concordance class of a link in the 3-sphere is uniquely determined by its Alexander polynomial for 2-component links with Alexander polynomial one. A similar result for knots with Alexander polynomial…