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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What are the points of Spec(Vassiliev Invariants)?
Background
Recall that a (oriented) knot is a smoothly embedded circle $S^1$ in $\mathbb R^3$, up to some natural equivalence relation (which is not quite trivial to write down). The collection of ...
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Distinct knots with same $A$-polynomial
Are there two non-isotopic knots $K,K'$ in $S^3$ with the same $\mathrm{SL}_2(\mathbb C)$ $A$-polynomials? If it's an open problem, has anyone suggested a method for finding them, or a reason why no ...
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Maslov index and Heegard Floer homology
I am an undergraduate who wants to learn Knot Floer homology. I was told to start with this expository paper which was working quite well until I reached the actual definition of the differential. ...
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A question on (1,1) bridge Knot
Hi, everyone. I am interested in the complement of (1,1) bridge knot in a lens space, $S^{3}$. Is there one (1,1) bridge knot in $S^{3}$ or lens space such that its complement is
hyperbolic?
Note:...
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Classification of knots by geometrization theorem
I read this interview with Ian Agol, where he says:
"...I learned that Thurston’s geometrization theorem allowed a complete and practical classification of knots."
My question is:
How does this ...
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Is there a known method for finding the minimum bridge index of a knot?
It is easy to establish an upper bound $n$ for the bridge index of a knot by producing a diagram with the knot in $n$-bridge position.
Is there a known method to produce a reasonable lower bound ...
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Minimal diagrams of equivalent knots and type III Reidemeister moves
Knot theory is not my area so sorry if this is a trivially true or trivially false question. Given equivalent knots K and L and minimal diagrams D(K) and D(L) of K and L, respectively, is it always ...
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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 ...
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Kirby's theorem for 4-manifolds
In dimension 3, we have the celebrated Kirby theorem: Let $L_1, L_2$ be two links in the 3-sphere $S^3$; then they surgeries along them produce homeomorphic 3-manifolds if and only if they are related ...
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Inverse Kirby knot
Given an (oriented framed) knot $K$ in the 3-sphere $S^3$, we can perform a surgery along $K$ to get another 3-manifold $M$. From $M$, we can perform the inverse surgery back to $S^3$.
However, the ...
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Determining loops of knots
For smooth knots in $\mathbb R^3$ from the work of Waldhausen [On Irreducible 3-Manifolds Which are Sufficiently Large, Annals of Mathematics (2)
87.1 (1968), 56–88] it follows that the knot group ...
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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 $H^1(M; \mathbb{Z}/2\mathbb{Z})$...
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Why is the thing dual to a "meridian" called a "longitude"?
A pair of distinguished generators of the fundamental group $\pi_1(\partial(S^3 \setminus K))$ of the boundary torus of a knot complement are usually called the "meridian" and "...
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Revisiting Gordon-Luecke theorem
$\DeclareMathOperator\Diff{Diff}\DeclareMathOperator\GL{GL}$Here is an proof-sketch of a strengthened Gordon-Luecke theorem. This is presumably known, but is it written down somewhere? I am also ...
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What is (explicitly) known about the SL(n,C) character varieties of 3-manifolds?
The $SL(2,{\bf C})$ character variety of a 3-manifold with 1-cusp $M$ (like a knot complement in the 3-sphere) essentially coincide with the variety defined by the A-polynomial. Those polynomials are ...
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Normal form of framed links under Kirby moves
It's well known that any oriented closed 3-manifold (topological or smooth) can be obtained by surgerizing along a (framed oriented) link $L$ in the 3-sphere $S^{3}$. Even better, Kirby found a ...
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Computation of cusp shape from vertex invariants
Following Takahashi ("On the concrete construction of hyperbolic structure of 3-manifolds"), I was able to construct the Euclidean cusp cross-section for the 5_2 knot complement (please see ...
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Ambiguity in the unoriented knot connected sum
It is well known that there might be ambiguity in the unoriented knot connected sum if the knots concerned are not invertible.
E.g., consider 8_17, the only knot with crossing number 8 which is non-...
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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,...
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Physical Approach to Knot Categorification
Some recent work by Aganagic on knot categorification, Knot Categorification from Mirror Symmetry, Part II: Lagrangians, discusses two categorical approaches to categorification of quantum link ...
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"Effectivity" of braid notation for knots
Any knot or link can be written in braid notation (with implied closure of strands). Some natural questions:
Assume I don't allow inverses - only overcrossings are used as generators. Anything known ...
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Link invariants from modular categories (strictification and computation)
By the theory of Reshetikhin and Turaev, a modular tensor category $C$ gives rise to a link invariant. While $C$ is strict as a monoidal category (e.g. $\mathbb{Fib}$), calculating the link can be ...
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What are the order 5 symmetries of the the torus knot T(3,4)?
Generally, I am interested in understanding the free actions of finite cyclic groups on $S^3$ which leave invariant an oriented torus knot $T(p,q)$. For a specific example, consider the knot $T(3,4)$ ...
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Unknot recognition - how tangled does it get?
A recent algorithm unknots in quasipolynomial time. But I want to know what happens to the crossing number. Assuming your unknot has $n$ crossings, if I remember correctly it might be necessary to ...
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Minimal generators of the fundamental groups of Whitehead doubled knot
$\DeclareMathOperator\rank{rank}$My question involves a special kind of satellite knot named (Whitehead) doubled knot. The definition is quoted from Klassen's paper but the construction is pretty ...
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Minimal number of generators of satellite knot groups
In light of Knot groups with big number of generators, I was wondering...
Question 1 What is the minimal number of generators of the fundamental group of a satellite knot?
Another more specific ...
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Characteristic disks in $S^2 \times S^2$ for knots
I'm studying the article Genera and degrees of Torus Knots in $\mathbb{CP}^2$ and I ended up with a question.
We know that every knot is slice (i.e. bounds properly embedded smooth disk) in $S^2 \...
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Knot groups with big number of generators
I start by saying that I am not an expert in this field and I apologize if the question is too elementary.
Let $K$ be a knot in $S^3$. I denote by $\pi_1(K)$ the knot group, which is the fundamental ...
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Reference for birational equivalence of $A$-polynomial curve and character variety
For $K$ a knot in $S^3$, the character variety $\mathfrak{X}_K$ parametrizes conjugacy classes of representations $\pi_1(S^3 \setminus K) \to \operatorname{SL}_2(\mathbb C)$.
Another object that does ...
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Fáry-Milnor theorem for positively curved metrics on $S^3$?
I'm interested in generalizations the following well-known theorem of Fáry and Milnor.
Theorem. (Fáry-Milnor) If a simple closed curve $\gamma \subset \mathbb{R}^3$ is knotted, then the total ...
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Progress on the paper by Hass and Lagarias on Reidemeister moves needed for unknotting
Has any progress been made on the findings of 2001 paper by Joel Hass and Jeffrey C. Lagarias? The paper is
Joel Hass and Jeffrey C. Lagarias, The number of Reidemeister moves needed for unknotting, ...
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Thurston universe gates in knots: which invariant is it?
Today I discovered this nice video of a lecture by Thurston:
https://youtu.be/daplYX6Oshc
in which he explains how a knot can be turned into a "fabric for universes". For example, the unknot ...
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Which homotopy classes $S^3 \to S^2$ lift to embeddings $S^3 \to S^2 \times D^3$?
The question is, for a smooth embedding
$$f : S^3 \to S^2 \times D^3$$
one can compose the map $f$ with projection $\pi : S^2 \times D^3 \to S^2$, giving the map $\pi \circ f : S^3 \to S^2$.
Which ...
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How many different knot types can have the same shadow (projection)
If we take a knot diagram and ignore the over/under-crossing information, we obtain a shadow (or projection). Thus a shadow is simply a plane embedding of a 4-regular graph. Clearly, two non-...
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Möbius cross energy in $S^3$?
Let $\gamma_i$, $i=1,2$ be two loops in $\mathbb R^3$. The Möbius cross energy of the pair is defined by
$$
E(\gamma_1, \gamma_2)=\iint_{S^1\times S^1}\frac{|\gamma'_1(u)|\cdot|\gamma'_2(v)|}{|\...
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Presentations of mapping class groups in dimension $3$
For any closed oriented surface $M$, its mapping class group $MCG(M)$ can be generated by Dehn twists along certain curves on $M$. A presentation for the group $MCG(M)$ was found in [1] and then ...
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Link at infinity of a complex algebraic curve transverse to S^3 and non-singular in D^4
I am currently working on the following paper by Lee Rudolph: https://arxiv.org/abs/math/9307233
Using Kronheimer-Mrowka's theorem, he proves in page 6 that the slice Euler characteristic of a given ...
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Which knots appear as the singular locus of a polyhedral metric on the 3-sphere?
What can be said about a knot $K\subseteq S^3$ for which there exists a (Euclidean) polyhedral metric (aka Euclidean cone-manifold structure) on $S^3$ whose singular locus is precisely $K$? I'm ...
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Does an amphichiral knot admit roughly twice as many knot diagrams of a given crossing than a chiral knot?
Recall that a knot is amphichiral (or achiral) if there is a continuous deformation of the knot into its mirror image.
I'm interested in knowing when and whether we can use approaches like Stockmeyer ...
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Is there a combinatorial way to determine the coefficients of the universal finite-type invariant on a given knot?
There are various (equivalent?) descriptions of a universal finite-type knot invariant, e.g. https://arxiv.org/abs/q-alg/9603010. They take the form of formal power series valued in Feynman diagrams (...
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Is there a known Legendrian simple link?
Several knots like unknot, $4_1$, $3_1$ are known to be Legendrian simple, i.e., Thurston-Bennequin number and rotation number determine Legendrian type completely.
How about the same notion for link ...
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Notation for algebraic tangles
The Conway notation for rational knots is well known. I could use one for the more general algebraic tangles. As the name already says, one could build one out of tangle addition $+$, multiplication $*...
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The Kirby diagram of a manifold glued along the lens space $L(p,1)$
Suppose $K$ is a knot in $S^3$ with any framing and $m=m_0$ is its meridian with $-1$ framing. Suppose $m_1,\dots,m_{p-1}$ are unknots with framings $-2$, such that $m_{i-1}$ and $m_i$ are linked as a ...
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Computation of Colored HOMFLY Polynomials
I am trying to understand the colored HOMFLY polynomials. The theoretic description Anna Aiston gave in her PhD thesis is really nice, but what about the computation?
I would like to understand the ...
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Braids of fibered knots
There are some theorems saying that the closure of a braid of a specific form is fibered. For instance, a theorem of Stallings says that the closure of a homogeneous braid is a fibered knot.
I am ...
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Rack cohomology as derived functor cohomology
Let $X$ be a rack and $A$ be an $X$-module. By this paper, p. 33, we can associate a cochain complex $C^\bullet(X,A)$ to the pair $(X,A)$. This complex is explicitly defined by a differential $d$. I ...
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What is the determinant of the R-matrix defining the colored Jones polynomial?
Let $V_n$ be the $(n+1)$-dimensional irreducible representation of $\mathcal U = \mathcal{U}_q(\mathfrak{sl}_2)$, and let $\mathbf R \in \mathcal{U} \widehat \otimes \mathcal{U}$ be the universal $R$-...
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Space of all topological knots (tame and wild)
Does anyone know something about the space of all topological knots (injective continuous maps from $S^1$) in $\mathbb R^3$ (or in some manifold)?
In addition, what is known about wild knots?
I found ...
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Thompson's group F and algebraic links
There is a procedure, suggested by Vaughan Jones, which associates a link to every element of Thompson's group F. Also every knot or link in $\mathbb{R}^3$ can be obtained in this way. A subclass of ...
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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 ...
