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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Fundamental group of the space of smooth embeddings of $S^1$ into $\mathbb R^3$

Has the fundamental group of the space of smooth embeddings of $S^1$ into $\mathbb R^3$ been completely computed? Say the basepoint is an unknot. Maybe something is known for other components? If yes,...
Ivan Sergeev's user avatar
7 votes
2 answers
749 views

How do I efficiently find a sequence of Reidemeister moves between equivalent link diagrams?

In knot theory, two link diagrams are equivalent if and only if they can be related by performing a finite number of Reidemeister moves. But sometimes it is so confusing that I don't know which type ...
Idonknow's user avatar
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7 votes
1 answer
419 views

Small knots becoming isotopic after connect sum

I am interested in the following situation: I have two codimension-2 knots $K_1$ and $K_2$ in $S^n$ and they are not isotopic. Furthermore, $K_1$ is not isotopic to the mirror image of $K_2$ and ...
Sprotte's user avatar
  • 1,065
7 votes
3 answers
509 views

Is there a notion of "ribbon 2-category"?

It there some notion of ribbon 2-category, which would allow for, say, talking about the Seifert surface of links (which is a 1-morphism in some ribbon category) as a 2-morphism in the category? ...
Jakob's user avatar
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7 votes
1 answer
328 views

Can two fibered knots have the same exterior?

Suppose I have two distinct fibered knots in a homology sphere. Is it possible for them to have (orientation-preservingly) homeomorphic exteriors? See Oriented knot complement conjecture for ...
shestipalov's user avatar
  • 1,000
7 votes
2 answers
367 views

How does Thurston's Orbifold Geometrization imply that knots with meridional rank 2 are 2-bridge?

Problem 1.11 of Kirby's list asks whether every knot that has a knot group which can be generated by n meridians, but not less than n, is an n-bridge knot. There is a one-sentence update, saying that ...
chris c's user avatar
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7 votes
1 answer
829 views

Trace identities and the Kauffman Bracket skein module

Let's consider $K_t(M)$, the Kauffman bracket skein module (see this and this papers) of a three-manifold $M$. When $t=-1$, $K_t(M)$ is easily seen to be isomorphic to the ring of functions on the ...
John Pardon's user avatar
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Ideals in the ring of single-variable Laurent polynomials with integer coefficients

I'm writing some software to automatically compute things like Alexander modules, Alexander ideals, Milnor signatures and Farber-Levine pairings for 1 and 2-knot complements. So among other things I'...
Ryan Budney's user avatar
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7 votes
1 answer
290 views

Is it possible to separate two linked (geometric) circles in $\Bbb R^3$ by a set homeomorphic to the 2-sphere (with arbitrarily “bad” homeomorphism)?

$A$ and $B$ are two linked (geometric) circles in $\Bbb{R}^3$. (Let, for definiteness, both have radius = 1, the first lies in the $z=0$ plane and its center is the origin of coordinates $(0,0,0)$, ...
Mikhail Patrakeev's user avatar
7 votes
1 answer
435 views

Classification of knots in solid torus

What is known about the classification of knots in a solid torus $S^1 \times D^2$? Is enumerating them a reasonable problem? Do we get a similar classification as for knots in $S^3$? Ideally there ...
Calvin McPhail-Snyder's user avatar
7 votes
2 answers
294 views

A ribbon presentation for a torus knot

Let $K$ be a knot in $S^3$. It is well-known that the knot $K \# -\overline{K}$ is always ribbon. The following picture describes the connected sum of the left-handed torus knot $T(3,4)$ and the right-...
Max Schumann's user avatar
7 votes
1 answer
795 views

integer surgeries on knots

I have constructed a list of surgery coefficients which yield spherical space forms. For instance, there are two knots with different Alexander polynomials on which 29-surgery will give a small ...
anonymous's user avatar
7 votes
1 answer
910 views

Which knots' stick numbers are twice their crossing numbers?

Looking at a table of minimum stick numbers for knots (table here), it seems the known upper bound of $2 c(K)$ in terms of the knot crossing number $c(K)$ is realized by the trefoil $3_1$—it ...
Joseph O'Rourke's user avatar
7 votes
2 answers
402 views

Is every element of $Mod(S_{g,1})$ a composition of right handed Dehn twists?

Let $S_{g,1}$ be the surface of genus $g \geq 1$ and $1$ boundary component. Let $Mod(S_{g,1})$ be the mapping class group in which we allow isotopies to rotate the action on the boundary (...
Paul's user avatar
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7 votes
1 answer
312 views

What is the complexity of determining if a knot group is $\mathbb{Z}$?

It is known from the work of Waldhausen that the isomorphism problem for knot groups is decidable. What is then: The complexity of determining if a knot group is $\mathbb{Z}$? .i.e. same as the ...
Prathamesh's user avatar
7 votes
1 answer
452 views

Can we cut and rotate a particular region of a hyperbolic 3-manifold to get another (non-homeomorphic) hyperbolic 3-manifold?

I'm trying to learn more about hyperbolic 3-manifolds, in particular the geometric implications of doing hyperbolic Dehn surgery to suitable knot complements. Following this paper by Christian ...
asldjk's user avatar
  • 318
7 votes
2 answers
667 views

Does annular Khovanov homology detect the unknot (in annulus)?

Recently Kronheimer and Mrowka showed that Khovanov homology detects the unknot. It's still not known if the Jones polynomial detects the unknot. Does annular Khovanov homology detect the unknot in ...
Puraṭci Vinnani's user avatar
7 votes
2 answers
258 views

Can every large point set be connected to a given knot?

Let $K$ be a given knot, and $P$ a set of points in $\mathbb{R}^3$ in general position, general position in the sense that no three points are collinear and no four coplanar. Define the point-set ...
Joseph O'Rourke's user avatar
7 votes
2 answers
816 views

Are Fourier series of length 2 'asymmetric enough' to generate all crossing patterns? - A reformulation of the Fourier-(1,1,2) knot question

Given $N$ pairs of distinct real numbers $t_i, t'_i \in [0,1]$, $i = 1,\ldots,N$, we ask if there is a function $f(x) = \cos(2\pi mx+\alpha) + \gamma\cdot \cos(2\pi nx+\beta)$, with $m, n \in \mathbb{...
Christoph Lamm's user avatar
7 votes
1 answer
512 views

Resemblance between Birman-Murakami-Wenzl algebra representations and the Lawrence-Krammer representations

At the end of Stephen Bigelow's paper "Braid Groups are Linear", he mentioned that there is a striking resemblance between the matrices of the Lawrence-Krammer (LK) representations and "those of a ...
Elden Elmanto's user avatar
7 votes
1 answer
215 views

Singularities of PL embedding of surface in a contractible 4-manifold

I am trying to understand the article "A solution to a conjecture of Zeeman" by Akbulut, but I am not an expert in PL-geometry. As far as I understand, two statements should be true, but I ...
P. Tolo's user avatar
  • 674
7 votes
2 answers
239 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 ...
user8749's user avatar
7 votes
2 answers
687 views

Quantum E6/E7 knot polynomials

Has anybody seen seen quantum knot invariants associated to (E6, 27) or (E7, 56) worked out in the literature? Even for just simple knots like the trefoil or figure-8? I suspect these haven't been ...
Ross Elliot's user avatar
7 votes
2 answers
487 views

Computations of the Link homology categorifying the second colored Jones polynomial

Has anybody done computations of such a theory? Is there a place I could look up and see what the answers are for low crossing knots?
Charlie Frohman's user avatar
7 votes
1 answer
337 views

Decidability of knot equivalence in general 3-manifolds? Surface equivalence?

Given a closed orientable 3-manifold $M^3$ and two knots $K_1$ and $K_2$ in $M$, is there an algorithm to decide if $K_1$ and $K_2$ are isotopic? Is there an algorithm to decide if there is a ...
user101010's user avatar
  • 5,319
7 votes
1 answer
207 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 ...
Calvin McPhail-Snyder's user avatar
7 votes
1 answer
297 views

Assigning a "canonical geometry" to a Seifert surface

I originally posted this on stackexchange, but it hasn't gotten an answer. I hope it's not inappropriate for this forum. Suppose I have a knot $K: S^1 \hookrightarrow S^3$ with minimal genus Seifert ...
gdd's user avatar
  • 175
7 votes
1 answer
542 views

Milnor's isotopy invariant using spectral sequence?

I'm reading stalling's article "the augmented ideal in group ring" in Ann. Math. Studies 84(R. H. Fox memorial volume) In his final remark, he says that Milnor's link invariant could be interpreted ...
Topologieee's user avatar
7 votes
1 answer
770 views

What are these 3-manifolds from surgery?

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^...
QCD_IS_GOOD's user avatar
7 votes
1 answer
1k views

Closed formula for colored Jones polynomial of the trefoil? (reference request)

(EDIT: Powers of $q$ in the formula corrected.) I've been doing some computations with skein modules, and I found the following formula for the N-th colored Jones polynomial of the trefoil: $\frac{1}...
Peter Samuelson's user avatar
7 votes
1 answer
361 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} =...
maxematician's user avatar
7 votes
1 answer
376 views

Computing $\pi_2$ of the complement of a 2-knot or spaces with aspherical splittings?

As far as I know, it is an open question if the complement of a ribbon disk $D^2 \subset B^4$ is aspherical. In reading "Some remarks on a problem of J.H.C Whitehead" by Howie, it is noted that there ...
user101010's user avatar
  • 5,319
7 votes
0 answers
245 views

Generating cycles inside Tits' graph of words for a positive braid

Let $Br_n$ be the braid group and consider words in its generators (not in the inverses). Two such words define the same "positive" braid if one can be obtained from the other by commuting ...
Allen Knutson's user avatar
7 votes
0 answers
151 views

Does the non-cancelation theorem hold for 2-knots?

In Rolfsen's knots and links, he shows that, as a consequence of the unknotting theorem, that if you connect sum two knots and get the unknot, they both had to be unknotted. Does the same statement ...
Daniel H. Hartman's user avatar
7 votes
0 answers
125 views

Generalized Brunnian links

A Brunnian link of order $n$ is nontrivial link of $n$ rings that becomes a trivial link of $n-1$ rings if any ring is removed. They were classified up to link-homotopy by Milnor in 1954. This ...
Richard Stanley's user avatar
7 votes
0 answers
356 views

When do two knots have isomorphic fundamental bikeis?

A kei, also known as an involutive (or involutory) quandle, is a quandle $(Q,*)$ satisfying the involution condition that $(x*y)*y=x$ for all $x$ and $y$. Just like we can define a fundamental ...
Keshav Srinivasan's user avatar
7 votes
0 answers
265 views

Chromatic polynomial and the circle

In https://arxiv.org/pdf/1208.5781.pdf It is proved that there is spectral sequence converging to $H^*(M^G,R)$ with the E1 page given by the graph cohomology complex $C_A(G)$ where $A:=H^*(M,R)$. My ...
Matthew Levy's user avatar
7 votes
0 answers
206 views

IH-moves on trivalent graphs, and a complex that might be known to low-dimensional topologists

Here is a combinatorial problem which is hard to Google but seems like it might have a solution well known to people who study finite type invariants etc. Let $G_{g,b}$ denote the set of finite ...
Dan Petersen's user avatar
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7 votes
0 answers
160 views

Kinematics of rolling knots

It is well known that there are trefoil knots without tritangent planes, and with 3d printers one can print these beautiful objects and make them roll on planes. (An example:https://www.youtube.com/...
Pedro Namtior's user avatar
7 votes
0 answers
316 views

Stable, tame A-infinity isomorphism?

Linear duality provides a correspondence between differentials on free tensor algebras and A-infinity algebras. Under this correspondence, what is the A-infinity counterpart of "stable, tame ...
Eric Zaslow's user avatar
7 votes
0 answers
310 views

What is the historical connection between Zeeman's twist spinning and Fox's Examples?

Both Ralph Fox and (at that time, yet to be knighted) Sir Christopher Zeeman attended the 1961 Georgia topology conference. Fox's paper from that conference was his seminal work, "A Quick Trip through ...
Scott Carter's user avatar
  • 5,234
7 votes
0 answers
357 views

Is the category of tangles that includes, X, Y, and Lambda a free Frobenius braided category?

Consider the category whose objects are non-negative integers that are represented as dots along a line, and whose morphisms are generated by $X$---positive crossing, $\bar{X}$ --- negative crossing, $...
Scott Carter's user avatar
  • 5,234
6 votes
2 answers
1k views

Reference for a fact (?) on homeomorphic knot complements

Does somebody have a reference (or an argument why it should be true) for the following statement? “Let $K$ and $K'$ be knots in $S^3$. If there is an orientation-preserving homeomorphism $h : S^3 \...
Malte's user avatar
  • 63
6 votes
3 answers
807 views

Is a spin structure on a knot complement the same thing as an orientation of the knot?

There are a variety of characterizations of spin structures on the tangent bundle of a manifold. Two facts about them: Spin structures on $TM$ are an affine space over $H^1(M; \mathbb{Z}/2\mathbb{Z})$...
Calvin McPhail-Snyder's user avatar
6 votes
4 answers
562 views

Structure of foliations of codimension 2 on three dimensional torus

Is it possible to have a one-dimensional foliation on three dimensional torus such that the foliation has a trefoil knot as its leaf? Moreover, does a one dimensional foliation on three dimensional ...
Xifeng Su's user avatar
  • 173
6 votes
3 answers
514 views

Given a link $L\subset S^3$ how to construct a link $L'$ whose complement have hyperbolic structure?

Thurston claimed that almost all closed 3-manifolds are hyperbolic. To support this, he said that every closed 3-manifold is obtained by Dehn surgery along some link whose complement is hyperbolic. ...
Anubhav Mukherjee's user avatar
6 votes
2 answers
605 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
6 votes
4 answers
1k views

Generating ribbon diagrams for knots known to be ribbon knots

Is there a source in the literature for ribbon diagrams for the knot-table knots known to be ribbon knots? For example, I'm interested in doing a computation which needs as input a ribbon diagram for ...
Ryan Budney's user avatar
  • 42.8k
6 votes
3 answers
364 views

Surgery along an arc connecting the components of a $2$-component link gives the unknot

Math Overflow seems to have a dearth of low dimensional topology, but this seems like an interesting question. Let $L$ be a $2$-component link in $S^3$. Suppose that there is a framed arc joining the ...
Jim Conant's user avatar
  • 4,838
6 votes
1 answer
455 views

Satellite knot example

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