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 ...
Maarten Havinga's user avatar
-2 votes
0 answers
95 views

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....
Hauke Reddmann's user avatar
6 votes
1 answer
274 views

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 ...
pgadey's user avatar
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-4 votes
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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 ...
Tangle's user avatar
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5 votes
1 answer
132 views

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 ...
Judy_xyh's user avatar
2 votes
3 answers
564 views

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 ...
Craig Feinstein's user avatar
5 votes
1 answer
136 views

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 ...
Oblonski's user avatar
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6 votes
1 answer
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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 ...
Alessio Di Prisa's user avatar
2 votes
0 answers
130 views

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 ...
Hauke Reddmann's user avatar
6 votes
1 answer
291 views

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 $Y\times\{1\}$, $K$...
Qiuyu Ren's user avatar
3 votes
1 answer
219 views

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 ...
Hauke Reddmann's user avatar
4 votes
0 answers
139 views

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 ...
hopftype's user avatar
2 votes
0 answers
106 views

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 ...
Eric Ley's user avatar
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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/...
hopftype's user avatar
6 votes
1 answer
412 views

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| \...
David E Speyer's user avatar
1 vote
1 answer
110 views

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 (...
Hauke Reddmann's user avatar
1 vote
0 answers
53 views

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 ...
Pulcinella's user avatar
  • 5,565
2 votes
2 answers
188 views

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 ...
Alex Nho's user avatar
18 votes
1 answer
874 views

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 ...
Silvio Levy's user avatar
1 vote
0 answers
68 views

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 ...
Student's user avatar
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4 votes
1 answer
262 views

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 ...
Partha's user avatar
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175 views

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 ...
Claudio's user avatar
5 votes
0 answers
214 views

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 ...
Sidharth Ghoshal's user avatar
1 vote
0 answers
132 views

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 $...
0x11111's user avatar
  • 493
3 votes
1 answer
295 views

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 $...
seldom seen's user avatar
4 votes
0 answers
93 views

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 ...
Ivan So's user avatar
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6 votes
1 answer
321 views

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 ...
Andrea Marino's user avatar
6 votes
1 answer
387 views

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 ...
Nicholas James's user avatar
4 votes
0 answers
133 views

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 ...
Ian Agol's user avatar
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3 votes
0 answers
144 views

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 ...
Ian Agol's user avatar
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58 views

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 ...
Claudio's user avatar
3 votes
1 answer
317 views

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 ...
Nobilis's user avatar
  • 323
2 votes
1 answer
139 views

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 ...
bd99's user avatar
  • 23
2 votes
1 answer
213 views

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 ...
Anton Petrunin's user avatar
5 votes
1 answer
239 views

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 ...
Ktt's user avatar
  • 197
5 votes
1 answer
263 views

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 ...
ThorbenK's user avatar
  • 1,175
2 votes
0 answers
70 views

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 ...
Trebor's user avatar
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2 votes
1 answer
128 views

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\}\...
nim's user avatar
  • 357
8 votes
1 answer
327 views

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^* \...
Trebor's user avatar
  • 1,031
4 votes
2 answers
337 views

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 ...
João Lobo Fernandes's user avatar
8 votes
0 answers
216 views

$U_q(\mathfrak{g})$ 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 $\DeclareMathOperator\Rep{Rep}V\in \Rep U_q(\mathfrak{g})$ ...
Pulcinella's user avatar
  • 5,565
2 votes
0 answers
143 views

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 $\pi(K) = \pi_1(S^3 \setminus K)$...
Calvin McPhail-Snyder's user avatar
1 vote
0 answers
55 views

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 ...
knotMJ's user avatar
  • 71
5 votes
0 answers
89 views

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 ...
knotMJ's user avatar
  • 71
6 votes
2 answers
613 views

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 ...
Joe Boninger's user avatar
5 votes
2 answers
256 views

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 ...
Julian Chaidez's user avatar
2 votes
1 answer
132 views

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 ...
Nobilis's user avatar
  • 323
4 votes
2 answers
427 views

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 ...
I. S.'s user avatar
  • 41
1 vote
2 answers
196 views

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 ...
Mira T.'s user avatar
  • 11
3 votes
0 answers
99 views

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 ...
Sprotte's user avatar
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