# 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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### Borromean rings on $\Bbb{RP}^2$ and octonions

If I draw a trefoil knot on a projective plane and draw a circle around it touching the three outer parts of the curve. I can view this as a division of the projective plane in 8 triangles, viewed as ...

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### Generalized writhe

I think that the Stasiak 3D writhe can be seen as a semi-invariant (Reidemeister 2 and 3, but not 1, i.e. regular isotopy) taking integer values (if measured in 4/7 units), just as the ordinary writhe....

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### Loop manipulation subgroup of the braid group

Recently, I came across a subgroup of the braid group $B_{2n}$ that I'm calling the "loop manipulation" group $H_n$.
The idea is that we treat pairs of adjacent strands in the braid group as ...

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### Is 3d writhe for tight rational tangles quantized?

The 3d writhe of (simple) ideal (i.e. tight) knots is quantized, as Stasiak and colleagues have shown years ago.
What's the current state of progress on the question whether the writhe of tight ...

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### Is there a nontrivial ribbon knot concordance from a knot to itself?

It was conjectured by Gordon and recently proved by Agol that ribbon concordance defined a partial order on the semi group of knots. I know that this question is close related to the slice ribbon ...

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### Solving the unknotting problem by pulling both ends of the string

It is an open question as to whether there is a polynomial time algorithm for recognizing the unknot.
Consider the following procedure for doing so on an actual physical string: Suppose there is a ...

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### Heegaard splitting of figure-8 knot complement

It is well-known that the figure-8 knot complement in $S^3$ can be described as a circle fibration of a once-punctured torus. Is there also a description of the figure-8 knot as a Heegaard splitting ...

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### Knotted concordances of slice links

Are there any examples of a link $L$ such that:
$L$ is (strongly) slice, meaning that there exists a properly embedded collection $C$ of $n=|L|$ disjoint annuli in $S^3\times [0,1]$ such that $C\cap ...

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### Mutants or not?

Two 4-tangles (drawn unneccesarily complicated to show how they are
related - both are 6-tangles capped off with the same cap):
(alternate version with ends at the same point)
If I could turn over ...

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### Slice knots in 3-manifolds

Is there a nonslice knot $K\subset S^3$ that is slice in some closed oriented $3$-manifold $Y$? Here, when we say $K$ is slice in $Y$, it means that when regarded as a local knot in $Y\times\{1\}$, $K$...

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### Rational 4-tangles vs rational knots

The closure of a rational $4$-tangle is a rational knot. But is the converse true? We could tangle up even the unknot to a hopeless mess before cutting it up, and we could cut it were it "hurts ...

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### Coloured Jones polynomial at 4th root of unity and Arf invariant

Looking at the link invariants of $\operatorname{SU}(2)$ Chern-Simons theory, if we take the coloured Jones polynomial of a knot K, say $J_N^K$ at fundamental representation $N=2$, then we get the ...

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### A cell complex constructed from singular knots

Let $\mathcal K_n$ be the set of all $n$-singular knots up to isotopy,i.e. an immersion of $S^1$ into $\mathbb R^3$ with $n$ transverse double points that is an embedding when restricted to the ...

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### Coloured Jones polynomial of the mirror image of a multicomponent link

This question has been reposted from MathStackExchange
It is well understood that the usual Jones polynomial of a knot or link can be related to the Jones polynomial of the mirror image of the knot/...

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### Relations between relations in the positive braid monoid

The positive braid monoid on $n$ strands is the monoid with generators $s_1$, $s_2$, ..., $s_{n-1}$ and relations
$$s_i s_{i+1} s_i = s_{i+1} s_i s_{i+1} \qquad s_i s_j = s_j s_i \text{for}\ |i-j| \...

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### History of knot enumeration tables

There is much arbitraryness in the Rolfsen (and later) tables.
Of course anyone would name $7_1$ to be the first knot with
$n=7$ crossings, but already my own "natural" ordering attempt
(...

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### Reshetikhin-Turaev invariants from extended 3d TQFTs

Attached to any object $V\in \mathcal{C}$ of a ribbon category $\mathcal{C}$, Reshetikhin and Turaev have defined knot invariants
$$\tau_V(K)\ \in\ \text{End}_{\mathcal{C}}(1_{\mathcal{C}})$$
for ...

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### Inverse of a smooth concordance of smooth knots

We say that a smooth concordance of smooth knots C' is inverse to C if the concatenation C•C' is smoothly isotopic to the trivial cylinder.
I wonder if there are any known ways of inverting smooth ...

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### ID needed for one mathematician in group photo

The photo below was taken at MSRI in 1984 and MSRI has asked me to try to find out (on behalf of Lou Kauffman, Sofia Lambropoulou and Martha Jones) the identity of the mathematician farthest left, in ...

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### Knot invariants in WZW CFT via Holographic Principle

In the physics literature the Holographic Principle relates
theories in the bulk and the theories in the asymptotic boundary.
While the bulk theory is the 3D Chern-Simons theory, the
corresponding ...

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### Casson's knot invariant

$\DeclareMathOperator\SU{SU}$Informally speaking, the Casson invariant counts half the number of conjugacy classes of representations of the fundamental group of a homology $3$-sphere $M$ into the ...

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### How does the extra rope length of this link/tangle scale with the inner triangle size?

The symmetric chiral link made of three long intertwined/linked/tangled flexible ropes of radius 1 shown in the figure, whose 6 ends all lie in a plane at spatial infinity and which are pulled ...

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### Are there examples of different knots with identical Jones polynomials and different Seifert Genus?

I had asked this question on math.stackexchange 2 days back but came up empty handed so I wanted to ask it here.
Are there known examples of $2$ non equivalent knots that have identical jones ...

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### Higher dimensional Seifert surfaces and link numbers of higher knots

In 3-manifold topology, the notion of Seifert surface is well known. It is then used to define link numbers of knots.
Question: Consider embedding $N^n \rightarrow M^{2n+1}$ of n-dimensional manifold $...

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### How many configurations of tubes are there?

Can $n$ disjoint lines in $\boldsymbol R^3$ be knotted? No... Let $X_n$ be the configuration space of $n$ disjoint lines in $\boldsymbol R^3$. It is not hard to see that $X_n$ is path connected: Let $...

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### KLO for operations over braids

KLO is a program that permits you to the do twistings, band operations over knots or Kirby diagram. However, I couldn't find a function on KLO that permits me to do the same thing over braids.
Is ...

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### Different flavours of Vassiliev Conjecture

There is something that puzzles me about "Vassiliev's Conjecture". I am sure I am missing some detail which is obvious to the community, since there are several tightly related kind of ...

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### How to use a Heegaard diagram to retrieve the original 3-manifold that it represents?

(Disclaimer: I apologize that this is an introductory question for a forum like MathOverflow, but I have run out of ideas and resources to understand how this works, and I don't know where else to ask ...

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### Equiangular piecewise circular knot projection

Inspired by this question: say that a circle (or line) $c$ is equiangular with circles/lines $a, b$ if the three circles/lines intersect in two points and $c$ makes equal angles with $a$ and $b$.
For ...

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### Equiangular spherical knot projection

For which knots is there a projection to the 2-sphere (with finitely many double points corresponding to crossings) so that the arcs of the projection are geodesic and the two undercrossing arcs make ...

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### Is the 3d writhe of ideal knots proportional to their smallest possible 2d writhe?

In a knot, the (two-dimensional) or 2d writhe is the sum of all positive crossings minus the sum of all negative crossings. The 2d writhe is always an integer. There is also, for each knot, a smallest ...

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### Straightforward reference on the unknotting number being a knot invariant

I'm writing a Master's thesis on knot invariants and I'm trying to chase down the original source that introduced the unknotting number and perhaps proved that it is a knot invariant. The texts I've ...

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### English version of a paper by Gusarov

I am looking for the english translation of the paper in russian Variations of knotted graphs, geometric technique of n-equivalence, St. Petersburg Math. J. 12-4 (2001) by Gusarov.
There is a .ps file ...

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### Wrapping a suitcase with large rotational symmetry

This is a follow-up question to Can I wrap a suitcase with hair ties.
Now we know that it is possible to wrap a suitcase with hair ties without tying them together,
but can you do it with large ...

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### Complement of plane curve and knot

In Libgober's paper Alexander polynomial of plane algebraic curves and cyclic multiple planes, Example 2 (p.850), Libgober claims that the complement to this curve (i.e. $x^2u=y^3$ relative to the ...

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### 0-surgery on a fibered hyperbolic ribbon knot

Does there exist a fibering hyperbolic ribbon knot such that the 0 surgery is exceptional? If so does there exist such an example where the result of 0-surgery is Seifert fibered?
I tried looking at ...

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### Naturality of ribbon category twists

Tortile Tensor Categories by Shum defines a twist to be a natural transformation $\theta : \operatorname{Id} \to \operatorname{Id}$ satisfying some axioms. However, wikipedia and nLab worded the ...

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### Space of the trivial long knot in the thickened surface

Let $F$ be a compact oriented surface and $x_0\in F$ a basepoint. Consider the set $\mathcal E=Emb_0(I,F\times I)$ of embeddings $\sigma\colon I\hookrightarrow F\times I$, $\sigma(\partial I)=\{x_0\}\...

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### Is there a notion of "knot category"?

Consider a rigid braided monoidal category, with braiding $\beta_{x,y} : x \otimes y \cong y \otimes x$, and every object has a dual such that $\epsilon_x : 1 \to a \otimes a^*, \bar\epsilon_x : a^* \...

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### Knot theory in handlebodies of arbitrary genus

It is well known that not all graphs embed on the plane (e.g. the graph $K_{3,3}$). However, every finite graph embeds into a surface of some genus. One can think of this procedure as starting with a ...

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### $U_q(\mathfrak{g})$ is to knot theory as $U_q(\hat{\mathfrak{g}})$ is to $?$

Let $\mathfrak{g}$ be a finite dimensional semisimple Lie algebra over the complex numbers, e.g. $\mathfrak{sl}_n$.
Then every representation $\DeclareMathOperator\Rep{Rep}V\in \Rep U_q(\mathfrak{g})$ ...

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### Can distinct meridians commute in a knot group?

Suppose I have a knot $K$ in $S^3$. Given a diagram $D$ of $K$ I get the Wirtinger presentation $\langle x_1, \dots, x_a \mid r_1, \dots, r_c\rangle$ of its knot group $\pi(K) = \pi_1(S^3 \setminus K)$...

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### Knotted Traveling Salesperson route

Let us consider fixed points in space, if we apply the well-known Traveling Salesperson Problem algorithm, we get the shortest route. It can give a nontrivial knot in the three-space. The question is ...

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### Skein tree depth on a minimal knot or link diagram

The skein tree depth is the maximum length of the shortest path from a leaf to the root of a skein tree, among all leaves. The skein tree depth of an oriented knot or a link $L$, denoted $td(L)$, is ...

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### Powers of meridians in knot groups

Given a (tame) knot $K \subset S^3$, let $t \in G = \pi_1(S^3 - K)$ be any meridian. The Wirtinger presentation shows that $\langle \langle t \rangle \rangle = G$, where the notation indicates the ...

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### Canonically representing the monodromy of a hyperbolic manifold fibered over $S^1$

Let $Y$ be a hyperbolic manifold that fibers over $S^1$, with fibration $\pi:Y \to S^1$ with fiber $\Sigma$. Thurston states that the monodromy $\phi:\Sigma \to
\Sigma$ of this projection is then ...

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### References for algorithms for computing knot invariants

I'm wondering if there's compiled literature on well-known algorithms and their bounds for computing various knot invariants (I'm writing a Master's thesis on the subject).
I can find individual ...

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### Finite application of one of Reidemeister moves on a knot diagram

It is known that given a knot diagram we can transform it into a trivial unknot diagram by a series of Reidemeister moves. The key word is "series".
Can we transform any knot diagram using a ...

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### Abelian covering of link complement

I'm considering finite index abelian (regular) covering of link complement:
$$ X \rightarrow S^3\setminus L$$
where $L$ is a minimally twisted chain link.
I'm interested in covering space. Can we ...

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### Explicit parameterizations of complicated unlinks?

I have a somewhat empirical question which I hope is still welcome here. I would like to know how to write down explicit parameterizations of "complicated unlinks", say with 2 or 10 ...