# 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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### 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) ...
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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### 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 $p$-...
239 views

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: ...
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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### String rewrite system for algebraic knots/links?

$\newcommand\over{\vert}\newcommand\rot{\mathopen<#1\mathclose>}$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?