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.
951 questions
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Fundamental group of the complement of a codimension two submanifold
Let $M$ denote an arbitrary closed, connected, n-dimensional manifold for $n\geq 4$. Does there always exist a closed (not necessarily connected!) codimension two submanifold $S \subset M$ such that $\...
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Tangle hypothesis and ribbon category
The tangle hypothesis, when specialized to ordinary framed tangles, says that the framed tangles form the free braided category with all duals (i.e. considered as a 3-category, all the 1- and 2-...
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Hat knot Floer Homology with Z coefficients calculation
I would like to ask for recommended references which carry out the calculation of the hat knot Floer homology of a knot with $\mathbb{Z}$ coefficients, i.e., $\widehat{\operatorname{HFK}}(K;\mathbb{Z})...
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When is a 2-bridge knot hyperbolic?
It is known that 2-bridge knots in $S^3$ can be classified by the Schubert form. My question is: which 2-bridge knots are hyperbolic? (Do we have a complete classification for hyperbolicity in 2-...
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Necessary condition for invertible knot concordance from both ends
It is clear that if $K_1$ and $K_2$ are two concordant knots by a concordance that only present ambient isotopic phenomena (no saddles, maxima, or minima) they are invertible concordant from both ends....
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Existence of orientable finite volume complete cusp hyperbolic 3-manifolds $\mathbb{H}^3 / \Gamma$, where $\Gamma$ has no parabolic generators?
Let $K$ be a hyperbolic knot, i.e., $S^3 - K$ is an orientable finite volume cusp hyperbolic 3 manifold. Let $M=S^3 - K$ then $M= \mathbb{H}^3/\Gamma$, where $\Gamma$ (Kleinian group) is discret ...
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Knots having the same Alexander module which are not S-equivalent
As is discussed in this answer, S-equivalence class can be regarded as the Alexander module plus Blachfield pairing, an isomorphism of some duals of Alexander modules.
There are examples of knots ...
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Slice-ribbon conjecture in other 3-manifolds
There is some notion of what it means for a knot $K\hookrightarrow M$ to be "slice." In particular, we may ask, for example, that there is a topologically embedded disk in $M\times[0,1]$ ...
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Finite two-relator groups and quotients of knot groups
Let $G$ be a one-relator group $\langle A \mid R = 1 \rangle$. Then clearly $G$ is finite if and only if it is cyclic of finite order, i.e. can be given by a presentation $\langle a \mid a^n = 1 \...
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Blackboard framing for embedded surfaces
A framed link is a link $L\subset\mathbb{R}^3$ equiped with the data of a normal vector field, called the framing; an isotopy of a framed link is an ambient isotopy, altering the framing accordingly. ...
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Asymptotic growth of twisted alexander polynomials and hyperbolic volume for infinite families of knots
Let $\{K_n\}_{n=1}^\infty$ be an infinite family of hyperbolic knots with increasing crossing number, and let $\rho_n: \pi_1(S^3 \setminus K_n) \to SL_N(\mathbb{C})$ be a sequence of irreducible ...
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Knot group of mirror image [closed]
Are the knot group and the knot group of its mirror image isomorphic?
And,How about the case of knotted surfaces?
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Finding an overtwisted disk in a contact surgery diagram
I've been reading through Surgery on Contact 3-Manifolds and Stein Surfaces by Ozbagci & Stipsicz and have been stuck on an exercise. Consider the following contact surgery diagram for $(S^3,\xi_{-...
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A uniform upper bound for the linking number of periodic orbits of algebraic vector fields
Inspired by these two posts on knots orbits of polynomial vector fields on $\mathbb{R}^3$(A polynomial vector field on $\mathbb{R}^3$ which has a knot periodic orbit) and (Are total curvature and the ...
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A polynomial vector field on $\mathbb{R}^3$ which has a knot periodic orbit
Is there a polynomial vector field
$$P(x,y,z)\partial_x+Q(x,y,z)\partial_y+R(x,y,z)\partial_z$$
which has a closed orbit $K$ such that $K$ is a non trivial knot?
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Seifert surfaces of fibered knots
Given a fibered knot $K\subseteq S^3$, does every genus-minimizing Seifert surface appear as the fiber of a bundle $S^3\setminus K\to S^1$?
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Does suspension preserve the inequivalence of knots?
Let $S$ be the suspension operator. Let $K1$ and $K_2$ be two knots in $S^3$ which are not equivalent. Does this imply that their suspensions in 4 sphere are not equivalent in the sense ...
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Are total curvature and the unknoting number of closed orbits of algebraic vector fields bounded uniformly by the degree of vector field?
I am interested in this question since 1999 when I heared the definition of a knot and I read the definition of unknoting and the total curvature of a knot.
To what extent can closed ...
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Horizontal knots on 3 sphere
Motivation: First I present my motivation for this question but this motivational part is not my main question.
I participated in a talk on knot theory. Then I presented the following ...
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Is there a "simplest" way to embed a graph in 3-space?
I consider embeddings of graphs into 3-space with edges embedded as arbitrary curves. In the simplest (non-trivial) case the graph $G$ is a cycle or union of cycles, in which case the embeddings can ...
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How to properly define a slice knot (or a locally flat disk)?
A knot $K\subset\Bbb S^3=\partial \Bbb D^4$ is said to be (topolopgically) slice if there is a locally flat disk $D\subset\Bbb D^4$ with $\partial D=D\cap \Bbb S^3=K$. As far as I understand, locally ...
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Is there a Dehn-like presentation of a knot quandle?
The knot group can be presented using either a Wirtinger presentation (with generators corresponding to arcs of the knot diagram) or a Dehn presentation (with generators corresponding to regions of ...
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Link invariants on a thickened surface
Let $\Sigma$ be an oriented surface. I want to know about link invariants in $\Sigma\times [0,1]$. I already know the Ozawa polynomial introduced in this paper, but I couldn’t find any other than that....
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link cobordism classes
Just an extremely basic question on link cobordism:
It would seem that all (framed) 1-links in $\mathbb{R}^3$ are (framed) cobordant to (framed) unlinks and in fact: to (framed) unknots --- for ...
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Cup-product ($\cup$) vs. cross-product ($\times$) on the space of graph homology
I'm currently reading Nieper-Wißkirchen's Chern numbers and Rozansky-Witten invariants of compact Hyper-Kähler manifolds. He introduces the space of graph homology $\mathcal B$ as the free $K[\circ]$-...
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Property P and R for general 3-manifolds
Let $Y$ be a closed oriented $3$-manifold and $K$ be a knot in $Y$. We say $K$ is the unknot if $K$ is contained in a local $3$-ball in $Y$ and is unknotted therein.
Generalized Property R:
If a Dehn ...
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Seifert invariants for Brieskorn manifolds $\Sigma(p,q,r)$
I've been studying Brieskorn manifolds $\Sigma(p,q,r)$ for $p,q,r\geq 2$. I know they are defined as the intersection of the complex surface $z_1^p+z_2^q+z_3^r=0$ as a subset of $\mathbb{C}^3$ and $S^...
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If the complement of a knot $K$ fibers over the circle is $K$ necessarily fibered?
Let $K \subseteq S^3$ be a knot in the $3$-sphere and assume there exists a smooth map $p \colon S^3\setminus K \to S^1$ which is a fiber bundle.
For every point $\require{enclose} \enclose{...
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Numerical computation of the second Vassiliev invariant, and the permutation $(1 3 4 2)$
$\DeclareMathOperator\SLL{SLL}$For a smooth embedding $\gamma(t):\mathbb{S}^1\rightarrow\mathbb{R}^3$, the Vassilev invariant of degree 2, which I will denote as $\nu_2(\gamma)$, may be computed ...
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Are isomorphisms of fundamental quandles canonical?
Given an (oriented) knot projection, we can create a quandle by assigning each curve segment a generator, and a relation $a \triangleright b = c$ for each intersection. If two projections are related ...
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Power series expansions and limits of knot invariants
This question is moved from math stackexchange which I posted several days ago without an answer.
Background(ignore this paragraph if you know finite type invariants well): Recall that a finite type ...
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existence of triangulations of manifolds
Let $M$ be a smooth manifold.
Let $K$ be a simplicial complex.
Let ${\rm sd}(K)$ be the sub-division of $K$.
Suppose there exists a simplicial sub-complex $K_1$ of ${\rm sd}(K)$ such that $K_1$ ...
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$3$-manifold that is a surgery on a knot
By the Lickorish-Wallace theorem, every oriented closed $3$-manifold can be obtained by a surgery on a link in $S^3$. In the statement of this result, links are required: not every such manifold can ...
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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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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 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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