Related papers: Long line knots

Bott, Cattaneo and Rossi defined invariants of long knots $\mathbb R^n \hookrightarrow \mathbb R^{n+2}$ as combinations of configuration space integrals for $n$ odd $\geq 3$. Here, we give a more flexible definition of these invariants. Our…

Lagrangian embeddings $\mathrm{S}^1\times\mathrm{S}^{n-1}\hookrightarrow\mathbb{R}^{2n}$ are classified up to smooth isotopy for all $n\ge 3$.

We study the spaces of embeddings $S^m\hookrightarrow R^n$ and those of long embeddings $R^m\hookrightarrow R^n$, i.e. embeddings of a fixed behavior outside a compact set. More precisely we look at the homotopy fiber of the inclusion of…

This paper contains the first knot polynomials which can distinguish the orientations of classical knots and which make no excplicit use of the knot group. But they make extensive use of the meridian and of the longitude in a geometric way.…

The aim of this paper is to realise the techniques of picture-valued invariants and invariants valued in free groups for long knots in the full torus. Such knots and links are of a particular interest because of their relation to Legendrian…

We define relative versions of the classical invariants of Legendrian and transverse knots in contact 3-manifolds for knots that are homologous to a fixed reference knot. We show these invariants are well-defined and give some basic…

This manuscript introduces a new framework for the study of knots by exploring the neighborhood of knot embeddings in the space of simple open and closed curves in 3-space. The latter gives rise to a knotoid spectrum, which determines the…

A polynomial knot in $\mathbb{R}^n$ is a smooth embedding of $\mathbb{R}$ in $\mathbb{R}^n$ such that the component functions are real polynomials. In the earlier paper with Mishra, we have studied the space $\mathcal{P}$ of polynomial…

We work in the smooth category. Let $N$ be a closed connected orientable 4-manifold with torsion free $H_1$, where $H_q := H_q(N; \mathbb Z)$. Our main result is a readily calculable classification of embeddings $N\to\mathbb R^7$ up to…

Ropelength and embedding thickness are related measures of geometric complexity of classical knots and links in Euclidean space. In their recent work, Freedman and Krushkal posed a question regarding lower bounds for embedding thickness of…

We construct two complete invariants of oriented classical knots in space. The value of each invariant on any knot is a set, infinite for the first invariant and finite for the second. The finite set is computed algorithmically from any…

We initiate the study of classical knots through the homotopy class of the n-th evaluation map of the knot, which is the induced map on the compactified n-point configuration space. Sending a knot to its n-th evaluation map realizes the…

We classify rotary (orientably-regular) maps whose underlying graphs are multicycles. For the multicycle $\mathrm{C}_n^{(\lambda)}$ of length $n$ and edge-multiplicity $\lambda$, we determine all rotary embeddings for $n\geqslant 3$ and…

We refine the Polyak-Viro Gauss diagram formula for the Vassiliev invariant of order two in a very simple way for the 2-cable of a framed long knot. Surprisingly, the resulting isotopy invariant of framed knots can detect already the…

We define new invariants of knots by means of quandle colorings and longitudinal information. These invariants can be applied to a tangle embedding problem and recognizing non-classical virtual knots.

Let $M_1$ and $M_2$ be closed connected orientable $3$-manifolds. We classify the sets of smooth and piecewise linear isotopy classes of embeddings $M_1\sqcup M_2\rightarrow S^6$.

Using convex surfaces and Kanda's classification theorem, we classify Legendrian isotopy classes of Legendrian linear curves in all tight contact structures on $T^3$. Some of the knot types considered in this article provide new examples of…

We present two different constructions of invariants for Legendrian knots in the standard contact space $\R^3$. These invariants are defined combinatorially, in terms of certain planar projections, and are useful in distinguishing…

A polynomial knot is a smooth embedding $\kappa: \real \to \real^n$ whose components are polynomials. The case $n = 3$ is of particular interest. It is both an object of real algebraic geometry as well as being an open ended topological…

Finite-order invariants of knots in arbitrary 3-manifolds (including non-orientable ones) are constructed and studied by methods of the topology of discriminant sets. Obstructions to the integrability of admissible weight systems to…