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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Is there any genus one knot other than $5_2$ with Alexander polynomial $2t^2-3t+2$?
It is known that genus one fibred knots are two trefoils and the figure-eight knot. Is there any characterization of the knot $5_2$? Specifically, is there any other genus one knot that shares the ...
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Is there a geometric interpretation of the second derivative of the Alexander polynomial at $1$?
For an (oriented) knot in $S^3$ the number $\Gamma(K) := \Delta_K’’(1)$ shows up in a number of places in knot theory, for example the Casson-Walker-Lescop invariant. Here $\Delta_K(t)$ is the ...
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Smoothing tame topological knots, from an analytic perspective
A tame topological knot (for the purpose of this question) is a topological embedding of $S^1 \times D^2$ into $\Bbb R^3$.
Tame topological knots are known to be isotopic to smooth knots. This ...
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Poincaré dual of the Alexander dual of the fundamental class of a knot is given by a Seifert surface
Let $K\subset S^3$ be an oriented knot and let $F:\overline{B^2}\times K\rightarrow S^3$ be a thickening with self linking number $0$. I will denote $F(B^2\times K)$ by $(B^2\times K)$ for simplicity. ...
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Surgery along knots and connected sum
Denote $S^3_{p/q}(K)$ by performing $p/q$-surgery along a knot $K$ in $S^3$.
Let $K$ and $J$ be two arbitrary oriented non-trivial knots in $S^3$. Is there a nice relation between surgery on the ...
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Isomorphism type of the knot concordance group
Isotopy classes of oriented knots in $S^3$ form a commutative monoid with respect to connected sum. Smoothly slice knots, i.e. knots that are the boundary of a smooth properly embedded disk in $B^4$, ...
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The knot $K\subset \Bbb S^3$ is smoothly slice, but the disc $D\subset \Bbb D^4$ is only locally flat. Can $D$ be smoothed?
Suppose I am given a smoothly slice knot $K\subset\Bbb S^3$. But I am only given a locally flat disc $D\subset \Bbb D^4$ with boundary $K$.
Question: Is there a smooth disc $D'\subset\Bbb D^4$ with ...
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A (possible) generalization of the unknot, inverses and the knot concordance
The notion of a knot concordance is a rich subject in low-dimensional topology, see Livingston's survey. More precisely:
For $i=0,1$, let $K_i$ be knots in $S^3$. A knot concordance from $K_0$ to $K_1$...
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Motivating quantum groups from knot invariants
Quantum groups are useful for making knot/link invariants: for example, $U_q(\mathfrak{sl}_2$) you get the Jones polynomial. This boils down to the fact that $\mathcal C = \operatorname{rep }U_q(\...
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What is known about exceptional slopes of hyperbolic knots?
For $K$ a hyperbolic knot in $S^3$, a rational number $p/q$ is an exceptional slope if the manifold $M_{p/q}$ obtained from $(p,q)$-Dehn surgery on $K$ does not admit a hyperbolic structure.
Thurston ...
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An equivalence relation on knots similar to concordance
Let $L_1$ and $L_2$ be two nonintersecting picewise-linear or smooth knots in $\mathbb R^3$. Suppose they are ambient isotopic. Does there exist an embedded surface $f: S^1\times[0,1]\to \mathbb R^3$ ...
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Do knot/link Floer homology detect variations of link genus?
$\newcommand{\wHFK}{\widehat{\mathrm{HFK}}}\newcommand{\wHFL}{\widehat{\mathrm{HFL}}}$Ni has shown that the knot Floer homology $\wHFK$ of an oriented link $L$ (in $S^3$ or more generally homology 3-...
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A knot in the solid torus and a Mazur manifold
Part 1: The following picture is from Saveliev's book Lectures on Topology of 3-manifolds, page 130:
He indicates that the knot drawn in the solid torus $S^1 \times D^2$ is homologous to $S^1 \times \...
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Natural knot homology
All knot homology theories I've seen share a flaw: their definitions explicitly use some combinatorial choices (such as a diagram presentation). The coin, however, has two sides and the other one is ...
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Gordon's approach: slice knots and contractible $4$-manifolds
Let $K \subset S^3$ be a slice knot. Then it bounds a smooth embedded disk $D \subset B^4$. Let $S^3_{p/q}(K)$ denote a $3$-manifold obtained by $p/q$-surgery on $K \subset S^3$.
The following theorem ...
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Non-commutative knot invariants
$\newcommand{\ab}{\mathrm{ab}}$Let $L=K_1\cup \dots \cup K_r$ be a link embedded in a 3-sphere. Here, $K_1,\dots, K_r$ are the component knots of $L$. A prototypical invariant associated with $L$ is ...
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Different 2D projections of a 3D knot
Somewhat relevant
Just curious: Obviously two projections of an embedding $S^1 \to \Bbb R^3$ can differ by a Reidemeister move. Or more of them. But when can two diagrams for the same knot (or link) ...
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Computation of $\pi_1$ for a Mazur manifold and its boundary
If we attach a $4$-dimensional $1$-handle $D^1 \times D^3$ to a $4$-dimensional $0$-handle $B^4$, we obtain $S^1 \times D^3$. The null homologous knot in $S^1 \times S^2$ indicated in the picture ...
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Double ($p$-fold) coverings of $B^4$ along ribbon/slice disks
I have two questions that seem to be related.
I wonder if there is a user-friendly algorithm (starting from ribbon/slice presentation of knots/disks) for the construction of double (in general $p$-...
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Quadrisecants of knots
Recall that a quadrisecant of a knot is a line that passes thru four points on it.
If the points appear on the line in the order $a$, $b$, $c$, $d$ and on the knot in the order $a$, $c$, $b$, $d$, ...
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Glueing two 2-tangles
Given 2-tangles $T_1,T_2\subset B^3$ with their endpoints at some fixed points NW, NE, SW, SE of $\partial B^3$ we can glue them along $\partial B^3$ to obtain a link $L=T_1\cup T_2\subset S^3.$
Q: ...
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Integral surgeries on $3$-manifolds
Let $K$ be a knot in $S^3$. Let $N(K)$ be a tubular neighborhood of $K$, a solid torus. On $\partial N(K)$, we may specify a preferred longitude $\lambda$, i.e., a simple closed curve whose linking ...
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String rewrite system for algebraic knots/links?
$\newcommand\over{\vert}\newcommand\rot[1]{\mathopen<#1\mathclose>}$By its definition, an algebraic tangle, and by extension, its closure (knot or link) can be written as a string (of ...
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Coxeter's braid group quotients
Coxeter's result is that if the generators of the braid group $B_n$ on $n$ strands fulfill a relation $\forall_i\sigma_i^k=1$, then $1/n+1/k>1/2$ must hold to get a finite quotient of $B_n$. In ...
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Tangles whose unknottedness realize boolean functions?
Let $f : \{0,1\}^n \to \{0,1\}$ be a Boolean function. Denote the two possible simple single crossing tangles by $T_0$ and $T_1$ (your choice for which is which). Is there some "generalized $n$-...
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Knots: locally flat, PL and smooth
In the classical dimension (knots in $S^3$), it is considered standard (I think?) that the following sets are in bijective correspondence:
locally flat knots up to ambient isotopy;
PL-knots up to PL ...
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Infinite family of different prime knots with trivial Alexander polynomial
I am looking for infinite families of prime knots that have all Alexander polynomial equals to 1. I wrote "families" (and not "family") since perhaps there are different ...
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Signature and cusp geometry of hyperbolic knots
Nature recently published a paper titled “Advancing mathematics by guiding human intuition with AI”. Using the power of linear algebra and calculus machine learning, the authors link "geometric&...
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Conjectures inspired by AI
Today in Nature a paper described how AI guided mathematicians to make highly non-trivial conjectures, which they managed to prove, one in Knot Theory involving a new invariant, the other in ...
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Knots with everywhere positive curvature
A naive question that my searches have not resolved:
Q. Can every knot be realized by a curve in $\mathbb{R}^3$ with strictly positive
curvature at every point?
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Deep learning for knot theory. Classification
As far as I know, there is a classification of all prime knots with less than 16 crossings.
It seems that there is already a fast enough algorithm to distinguish a knot from an unknot.
So in principle ...
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Hyperbolic volume of hyperbolic knots
Let $G$ be a torsionfree kleinian group. Is there necessary an sufficient conditions on $G$ to be a knot group ?
It seems that there is some necessary conditions:
$H_{1}(BG) = \mathbb{Z}$
$H_{2}(BG) ...
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Relative strength of Jones and colored Jones polynomials
this is my first post here.
I've been studying some Knot Theory and I came to a question concerning invariants.
We know that the Jones polynomial is related to the RT-invariant associated to the two-...
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Is this drawing of $K_{4,4}$ knotted?
Let $A$ and $B$ be skew lines in $\mathbb{R}^3$. Choose four points $a_1, a_2, a_3, a_4$ on $A$ and four points $b_1, b_2, b_3, b_4$. For all $i,j \in [4]$ draw a line segment from $a_i$ to $b_j$. ...
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Relation between symmetries of hyperbolic knot and the symmetries of a generic triangulation
For canonical ideal triangulation of a hyperbolic knot, the symmetries of the knot are the same as the symmetries of that triangulation. This is how SnapPy computes the knot symmetry group.
Is there ...
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Formula for the Casson invariant in terms of the linking form
The paper 'Trisections, intersection forms and the Torelli group' by Peter Lambert-Cole quotes the following formula for the Casson invariant of a knot $K$ in a homology $3$-sphere in terms of the ...
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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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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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Is the number of prime factors of 3-manifolds obtained by Dehn surgery along a link with $N$ components in $S^3$ bounded from above?
For a given $N$, is the number of prime factors of 3-manifolds obtained by Dehn surgery along a link with $N$ components in $S^3$ bounded from above?
The Two Summands Conjecture states that surgery ...
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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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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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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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Quadratic cusp shape
Which hyperbolic $3$-manifolds are known to have quadratic cusp shape?
Explanations: Cusps of hyperbolic $3$-manifolds are products torus x interval. They lift to horoballs in hyperbolic $3$-space, ...
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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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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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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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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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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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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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