# Questions tagged [knot-theory]

Knot theory is dealing with embedding of curves in manifolds of dimension 3. A knot is a single circle embedded in the affine space of dimension 3 as a smooth curve not crossing itself. Many knot invariants are known and can be used to distinguish knots.

748 questions
Filter by
Sorted by
Tagged with
85 views

### Is there a combinatorial way to determine the coefficients of the universal finite-type invariant on a given knot?

There are various (equivalent?) descriptions of a universal finite-type knot invariant, e.g. https://arxiv.org/abs/q-alg/9603010. They take the form of formal power series valued in Feynman diagrams (...
40 views

101 views

### Rolling wheel unicycle knots

Let $K$ be a knot, and $K(t)$ a parametrization of a space curve that realizes $K$. Roll a wheel $W$ of radius $r$ on $K(t)$ so that $W$ remains in the tangent-normal plane. Now track the wheel's ...
208 views

### Are knot invariants topological invariants? [closed]

I am a bit confused about terminology considering topology and knot theory. A topological invariant is considered to be a topological property that does not change under a homeomorphism of the space. ...
116 views

### Skein relation, Braids, and Hecke algebra

Many knot invariants (e.g. Alexander polynomial, Jones Polynomial,etc) admit a recursive algorithm based on the so-called skein relation But why the skein relation is a natural thing? People have been ...
142 views

### Questions about a few terminologies in “Knots and Links” by Rolfsen

In "Knots and Links" by Rolfsen, he mentioned words like *"the collar of a boundary", "bicollared boundary", "a bicollar on the boundary". I just wonder what ...
95 views

### Distinguishable knots (with constraints) over polyhedra

I'm trying to find the number of distinguishable knot projections over certain convex regular polyhedra according to the following constraints. On each face on the polyhedron the knot will have a ...
136 views

### Surveys on unknotting number

Any knot diagram could be converted to an unknot by cross change. The unknotting number of a knot diagram is the minimal number of cross changes needed. A knot could have many different diagrams and ...
156 views

### Can the Chern-Simons invariant of a cusped hyperbolic $3$-manfiold be defined mod $\mathbb Z$?

For a closed hyperbolic $3$-manifold $M$, the Chern-Simons invariant $CS(M)$ can be defined as an element of $\mathbb R/\mathbb Z$. When $M$ is cusped it can still be defined, but is now only defined ...
60 views

190 views

Let $H$ be a contact handlebody. In other words, $H$ is a small regular neighborhood of a Legendrian graph in a contact $3$-manifold (wlog $\mathbb R^3$). Equivalently, $H=(\Sigma\times[0,1],dt+\... 0answers 149 views ### references on categorification of knot invariants I am extremely sorry if this is not the right place for this kind of question. I have studied some knot theory, quantum invariants and would like to study more about categorification of knot ... 2answers 202 views ### What are the “correct” references for the Vassiliev invariant? Is there a good survey paper which describes the general ideas of Vassiliev's invariant? I am not an expert on knot theory, many references are too technical for me. Could Vassiliev's invariants be ... 0answers 89 views ### Is every tricolourable knot chiral? In other words, can anyone give an example of a tricolourable knot that is equivalent under the Reidemeister moves to its mirror image? If not, is it known or conjectured anywhere in the literature ... 5answers 386 views ###$0$-surgeries on trefoil and figure-eight Let$M$and$N$be$3$-manifolds obtained by zero-surgery on (left-handed) trefoil and figure-eight knot respectively. What is the easy way to prove that$M$and$N$are not homeomorphic? Note: When ... 0answers 80 views ### JSJ-type decompositions for knots According to Wikipedia, JSJ decomposition for 3-manifolds is the following statement: "Irreducible orientable closed (i.e., compact and without boundary) 3-manifolds have a unique (up to isotopy) ... 1answer 281 views ### Harmonic functions on knot complements In Axler's Harmonic Function Theory, he and his coauthors develop the theory of harmonic functions on spheres and discs by considering the restrictions of arbitrary polynomials on the sphere$S^{n-1} =...
Write the 4-ball as $\mathbb{D}^4=\mathbb{D}^2\times \mathbb{D}^2$. Then its boundary $\mathbb{S}^3\simeq \mathbb{S}^1\times \mathbb{D}^2\cup \mathbb{D}^2\times \mathbb{S}^1$. We will use implicitely ...