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# 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.

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149 views

### Tracking down an elusive book

A few weeks ago I had a very engaging talk with a faculty member, where he told me lots of interesting things about quantum algebras, know theory and Reshetikhin-Turaev invariants (this field is not ...
129 views

### Distinguishing Square Knot and Granny Knot using Quandles

It is known that the square knot and the granny knot are nonequivalent although they have isomorphic fundamental groups. I want to write a work on knot theory and provide these knots as an example ...
149 views

### Reference request: A knot is tame if and only if it has a tubular neighbourhood

Definitions: A knot is an embedding $\kappa:S^1\hookrightarrow S^3$ (we do not require smooth or polygonal). Two knots $\kappa,\,\lambda:S^1\hookrightarrow S^3$ are equivalent if one of the following ...
193 views

### Why does the longitude correspond to Frobenius in Arithmetic Topology, and other strange phenomena

I am trying to adress Morishita's book Knots and Primes to discover a bit about Arithmetic Topology, but some analogies puzzle me. I know that the correspondence should be addressed with a grain of ...
65 views

### Finding a presentation matrix with low dimension

Let $R=\mathbb Z[t^{\pm}]$ and $M$ a finitely generated $R$-module. With $A$ a presentation matrix, i.e we have the following exact sequence (usually I'm working with the case where $A$ is an square ...
132 views

### Are Turaev-Viro invariants holonomic?

Consider a 3-manifold $M$ with a boundary, which is a genus $g\geq 1$ surface $\Sigma$. Fix a triangulation $T$ of $\Sigma$. Then Turaev-Viro invariants $TV_q(M)$ are functions, assigning to integer ...
91 views

I was surprised recently to come across a 3-component link where the linking number of two of the components was negative. For a while I thought I had made a mistake, then I thought a little more and ...
109 views

### The bridge index and crookedness of a knot

I am reading Dale Rolfsen's book KNOTS AND LINKS, at page 115, I can't figure out why the crookedness of a knot equals its bridge index. Please give me some hints or any references available, much ...
147 views

I know that surgery on the unlink with +0 slope gives $S^2 \times S^1$ (where all the links above are embedded in $S^3$). I figured (I think) that surgery on the hopf link (with +0 on both) returns $S^... 2answers 133 views ### Does a knot and a tunnel exterior having free fundamental group imply it's an unknotting tunnel? The title is just about it. Assume we have a nontrivial knot$K$in$S^3$and the exterior of$K$,$E(K)$, is$S^3 \setminus N(K)$. Here$N(K)$is a regular neighborhood. Let$\tau$be a properly ... 1answer 211 views ### Sliceness of knots For a subring$R? \mathbb Q$, a knot$K?S^3$is called$R$-slice if there exists an embedded disk$D$in an$R$-homology$4$-ball$B$such that$?(B,D) = (S^3,K)$, see [Definition 1.3, KW16]. We say$...
167 views

(This is a follow-up question; the original question was about shelves.) A rack $(R, \rhd, \lhd)$ is a set $R$ with two binary operations $\rhd$ and $\lhd$ such that for all $x, y, z \in R$: $x \rhd ... 1answer 326 views ### Shelves with “trichotomy” A left shelf$(S, \rhd)$is a magma with the left self-distributive law: $$\forall x, y, z \in S: x \rhd (y \rhd z) = (x \rhd y) \rhd (x \rhd z).$$ Shelves are generalization of racks and quandles ... 1answer 66 views ### Genus of the surface traced out by a knot I am no good at visualizing things, and to add to the misery, have only a passing acquaintance with knot theory, so at the risk of sounding silly I dare ask in loose terms: if I take a knot$K$in$\...
152 views

### Chirality and Anti-Chirality of links in 3 and in 5 dimensions

We know there is a chiral knot which is a knot that is not equivalent to its mirror image. It is well known in the mathematical field of knot theory: https://en.wikipedia.org/wiki/Chiral_knot My ...

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