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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9 votes
1 answer
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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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6 votes
0 answers
110 views

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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4 votes
1 answer
157 views

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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8 votes
1 answer
261 views

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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8 votes
1 answer
118 views

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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10 votes
1 answer
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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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6 votes
1 answer
196 views

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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2 votes
1 answer
180 views

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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7 votes
2 answers
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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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2 votes
2 answers
156 views

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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5 votes
0 answers
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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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9 votes
2 answers
364 views

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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11 votes
0 answers
147 views

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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8 votes
2 answers
185 views

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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5 votes
1 answer
180 views

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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4 votes
0 answers
125 views

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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5 votes
1 answer
194 views

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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6 votes
2 answers
205 views

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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9 votes
1 answer
239 views

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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3 votes
2 answers
108 views

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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5 votes
1 answer
177 views

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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4 votes
0 answers
78 views

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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5 votes
0 answers
150 views

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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4 votes
0 answers
70 views

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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6 votes
1 answer
327 views

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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5 votes
3 answers
180 views

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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6 votes
0 answers
261 views

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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23 votes
1 answer
1k views

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 ...
0 votes
0 answers
97 views

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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9 votes
3 answers
799 views

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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3 votes
2 answers
227 views

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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3 votes
0 answers
84 views

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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10 votes
1 answer
282 views

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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1 vote
0 answers
40 views

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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3 votes
0 answers
96 views

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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8 votes
2 answers
218 views

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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7 votes
0 answers
122 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 ...
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4 votes
0 answers
200 views

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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8 votes
0 answers
126 views

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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2 votes
1 answer
132 views

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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9 votes
1 answer
677 views

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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  • 4,005
4 votes
1 answer
57 views

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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  • 4,005
4 votes
2 answers
198 views

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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11 votes
1 answer
386 views

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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  • 40.6k
4 votes
1 answer
178 views

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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  • 43
5 votes
3 answers
701 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 $H^1(M; \mathbb{Z}/2\mathbb{Z})$...
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3 votes
1 answer
334 views

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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1 vote
1 answer
110 views

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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  • 11
7 votes
2 answers
317 views

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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