A knot in S^3 is small if its complement does not contain a closed incompressible surface. Is it a generic property for knots, meaning that among all knots with less than $n$ cross …

Why do knot cobordisms result in functoriality with respect to knot homologies so often?

A topological ("closed") knot is an embedding of a circle in $\mathbb{R}^3$. It's possible for a knot to be distinct from the unknot because there are no free ends to move around a …

Let $B_n$ be the braid group on $n$ strands, with the usual generators: $s_1, \ldots, s_{n-1}$ and their inverses, where $s_i$ is a positive half-twist interchanging the strands la …

I know $\pi_4(S^2)$ is $\mathbb{Z}_2$. However, I don't know how to visualize it. For example, it is well known that $\pi_3(S^2)=\mathbb{Z}$ can be understood by Hopf Fibration. El …

Hello, i am very interested in knot theory, especially in knot groups and knot polynomials. Therefore i am reading the book of Crowell and Fox (Introduction to knot theory). I wa …

Consider three 2-torus ($S^1*S^1$) living in four space. Can I have links of these objects, which is generalization of links of circles in 3D? If so, how can I judge whether three …

Let $S$ be a sphere of unit radius. Let $C_n$ be a collection of unit-radius circles/rings whose centers are (uniformly distributed) random points in $S$, and which are oriented (t …

Does anyone know of a pair of different links which the HOMFLY polynomial does not distinguish, but HOMFLY homology does? Or does there exist such a pair of links? I'm assumi …

I am now facing a problem with "surface link" in four dimension. I have heard that three 2-torus can be linked in four dimension. And I have created a movie by cutting four dimensi …

Hello :) i have just reading the question "How to compute the Alexander polynomial of general torus knot" and i was suprised how strong it works if someone have a difficult questio …

Consider a knot to be a diagram in a plane--- i.e. a drawing of a finite connected planar graph (loops and multiple edges allowed) whose vertices are 4-valent with cyclic ordering …

The knot group is the fundamental group of the knot complement in $S^{3} $. The Dehn presentation of the knot group is a particular group presentation obtained by looking at the r …

It is well-known that there exist pseudo-Anosov automorphisms of surfaces that act trivially on the homology: they form the Torelli group. Similarly there exists pseudo-Anosov auto …

Can someone provide me with an example of a satellite knot with symmetry group which is neither cyclic nor dihedral?