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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Are there any very hard unknots?
Some years ago I took a long piece of string, tied it into a loop, and tried to twist it up into a tangle that I would find hard to untangle. No matter what I did, I could never cause the later me any ...
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can you fool SnapPea?
A while back I thought I had some simple knots that "fooled" SnapPea. But I no longer remember those examples, if I ever had them to begin with.
What I'm looking for is a non-hyperbolizable knot ...
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Number of the Reidemeister moves needed to transform one diagram into another one
A recent question Random Reidemeister moves to unknot contains a link to the paper http://www.ams.org/journals/jams/2001-14-02/S0894-0347-01-00358-7/S0894-0347-01-00358-7.pdf, in which J. Hass and J. ...
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Can knot diagrams be monotonically simplified using under moves?
It is well known that knot diagrams cannot be monotonically simplified using Reidemeister moves. For instance, the Goeritz unknot cannot be directly simplified. On the other hand, there is a stronger ...
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Can I wrap a suitcase with hair ties
Is there a nontrivial link in a big solid torus that is trivial in the ambient Euclidean space such that each circle is unknot and has a sufficiently small length?
It is motivated by a question that ...
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Applications of knot theory
An answer of André Henriques' inspired the following closely related CW question. Parts of the following is extracted from his answer and my comments.
I regularly teach a knot theory class. ...
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Proof of the Reidemeister theorem
While preparing for my introduction to topology course, I've realized that I don't know where to find a detailed proof of the Reidemeister theorem (two link diagrams give isotopic links, iff they can ...
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What is the knot associated to a prime?
I can't help but ask this question, having found out about arithmetic topology here on MO. There is a concise description of the MKR dictionary central to this philosophy here. This dictionary is used ...
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Complete knot invariant?
I've seen a couple papers (that I now can't find) that say that in his paper "On irreducible 3-manifolds which are sufficiently large" Waldhausen proved that the data $\pi_1(\partial (S^3\setminus K)) ...
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Complexity of random knot with vertices on sphere
Connect $n$ random points on a sphere in a cycle of
segments between succesive points:
I would like to know the growth rate, with respect to $n$, of the crossing number
(the minimal number of ...
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Several questions about Gauss's mathematical conception of braids
I'm trying to figure out several things about Gauss's thoughts concerning a certain four-strand braid. The reference my questions are based on is mainly Moritz Epple's excellent article "orbits ...
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What is the state of the art for algorithmic knot simplification?
Question: Given a `hard' diagram of a knot, with over a hundred crossings, what is the best algorithm and software tool to simplify it? Will it also simplify virtual knot diagrams, tangle diagrams, ...
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Random rings linked into one component?
Let $S$ be a sphere of unit radius.
Let $C_n$ be a collection of unit-radius circles/rings whose centers
are (uniformly distributed)
random points in $S$, and which are oriented (tilted) randomly (...
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Slice knots and exotic $\mathbb R^4$
In the http://arxiv.org/abs/math/0606464v1 I read
"If you want to prove existence of exotic smooth structure on $\mathbb R^4$ you can do this if you are in
possession of a knot which is ...
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Which convex bodies can be captured in a knot?
Which convex bodies can be captured in a knot?
This question is based on the discussion in "Is it possible to capture a sphere in a knot?".
We assume that the knot is made from ...
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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 a fact (?) on homeomorphic knot complements
Does somebody have a reference (or an argument why it should be true) for the following statement?
“Let $K$ and $K'$ be knots in $S^3$. If there is an orientation-preserving homeomorphism $h : S^3 \...
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Min Bend Orthogonal Knots
I am seeking literature on 3D orthogonal drawings of knots,
especially minimum bend drawings.
An orthogonal drawing employs segments parallel to the axes of
a Cartesian coordinate system.
A bend is a ...
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Tying knots with reflecting lightrays
Let a lightray bounce around inside a cube whose faces
are (internal) mirrors.
If its slopes are rational, it will eventually form a cycle.
For example, starting with a point $p_0$ in the interior
of ...
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Kirby calculus and local moves
Every orientable 3-manifold can be obtained from the 3-sphere by doing surgery along a framed link. Kirby's theorem says that the surgery along two framed links gives homeomorphic manifolds if and ...
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How to add essentially new knots to the universe?
A knot is an embedding of a circle $S^{1}$ in $3$-dimensional Euclidean space, $\mathbb{R}^3$. Knots are considered equivalent under ambient isotopy. There are two different types of knots, tame and ...
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Prime numbers as knots: Alexander polynomial
A naive and idle number theory question from a topologist (but not a knot theorist):
I have heard it said (and this came up recently at MO) that there is a fruitful analogy between Spec $\mathbb Z$ ...
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In knot theory: Benefits of working in $S^3$ instead of $\mathbb{R}^3$?
In several textbooks on knot theory (e.g. Lickorish's, Rolfsen's) knots are considered in $\mathbb{R}^3$ or $S^3$. The reason for working in $S^3$ is sometimes given at the beginning of a text as that ...
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Knot security (When to trust your life with a knot)
This question is related to a a question about self-tightening knots.
I am supervising a senior thesis and my student is interested in knots. My student is also a rock climber and has an ...
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Is there a "knot theory" for graphs?
I think knot theory has been studied for quite a while (like a century or so), so I'm just wondering whether there is a "knot theory" for graphs, i.e. the study of (topological properties of)...
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The Jones polynomial at specific values of $t$
I've been calculating some Jones polynomials lately and I was just curious if there was a "physical" (or, rather, geometric) meaning to evaluating the Jones polynomial at a particular value of $t$.
...
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Random knot on six vertices
This question is inspired by Joseph O'Rourke's beautiful question on random knots. Choose an random ordered 6-tuple of points on the unit sphere in $\mathbf{R}^3$, and form a knot by connecting ...
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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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How expensive is knowledge? Knots, Links, 3 and 4-manifold algorithms. [closed]
With geometrization, Rubinstein's 3-sphere recognition algorithm and the Manning algorithm, 3-manifold theory has reached a certain maturity where many questions are "readily" answerable about 3-...
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How many definitions are there of the Jones polynomial?
Even with the connection to quantum groups being made clearer (I believe it was not known when the Jones polynomial was first introduced), it seems to me that still we don't have the "right" ...
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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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Knots realized as algebraic curves
Two questions:
Q1. Have researchers worked out minimum-degree
real algebraic curves in $\mathbb{R}^3$ realizing specific knots?
Some work on the trefoil is reported in this MSE question.
&...
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Why is it so hard to implement Haken's Algorithm for knot theory?
Why is it so hard to implement Haken's Algorithm for recognizing whether a knot is unknotted? (Is there a computer implementation of this algorithm?)
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Define the 3d Chern-Simons TQFT on a discrete simplicial complex
Question: What is the challenge and the current status to define the 3d Chern-Simons(-Witten) (CSW) theory on a simplicial complex or on a discrete lattice? (Or is there a no-go or an obstruction ...
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On connection between Knot theory and Operator algebra
What exactly is the connection between knots and operator algebra? I heard that Jones established such a connection while discovering the celebrated Jones Polynomial.
Now Jones Polynomial is ...
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Can the n-string sphere braid group embed in to the (n+1)-string sphere braid group?
This question was originally posted on math.SE by myself nearly a year ago. I've been thinking again about the problem after it recently received a little attention, but little progress was made in ...
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Topological description of inverting a knot
The first figure shows an offset overhand knot. To tie it, you simply bring the two cords together and make an overhand knot in them. It's more secure than it looks, and several climbers have been ...
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What's about "quantum modular forms"?
Do you know where one could read on "Modular Forms, K-theory and Knots"? The combination of themes sounds thrilling!
Edit: Zagier's paper on "quantum modular forms" will be published in Clay's ...
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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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Human Knot game [duplicate]
In the popular game "Human Knot", a group of people stands in a circle and each person grabs another person's hands at random (one with the left hand and one with the right hand). The goal is to ...
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Link such that deleting any two components leaves an unlink
Brunnian links are well known, where deleting any component allows you to isotope the rest to an unlink. It's common to construct them by taking an $n-1$ component unlink and defining the $n$th ...
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HOMFLY and homology; also superalgebras
My understanding is that an analogy along the following lines is (roughly) true:
"The Alexander polynomial is to knot Floer homology is to gl(1|1)
as the Jones polynomial is to Khovanov homology is ...
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How many traces are there on Temperley-Lieb, Fuss-Catalan, Iwahori-Hecke, Birman-Wenzl-Murakami-Kauffman, ... algebras?
There is a theorem (I believe by Ocneanu) that the Markov trace on the tower of Temperley-Lieb algebras is (essentially) unique.
What about just traces on separate algebras? That is, take one of them,...
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A fun game related to knot theory
I recently learned the following rather fun game: a group of people is standing up roughly in circle, facing each other. Then participants randomly join hands, in such a way that nobody holds its own ...
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Descriptive Complexity of Knot Equivalence
I was reading a little about knots (in a popular math book that wasn't very good) and the book put forth several knot invariants like the Alexander and Jones polynomials. But these are not complete ...
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$0$-surgeries on trefoil and figure-eight
Let $M$ and $N$ be $3$-manifolds obtained by zero-surgery on (left-handed) trefoil and figure-eight knot respectively.
What is the easy way to prove that $M$ and $N$ are not homeomorphic?
Note: When ...
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Relating smooth concordance and homology cobordism via integral surgeries
Let $K_0$ and $ K_1$ be knots in $S^3$. They are called smoothly concordant if there is a smoothly properly embedded cylinder $S^1 \times [0,1]$ in $S^3 \times [0,1]$ such that $\partial (S^1 \times [...
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Knot theory question: bridge number vs. min generators of fundamental group of complement
Given a knot in the 3-sphere in Bridge Position you can find a presentation for the fundamental group of the complement (a Wirthinger presentation) containing $n$ generators and $n-1$ relators, where $...
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Is there a polynomial-time algorithm for untangling the unknot?
I've found assertions that recognising the unknot is NP (but not explicitly NP hard or NP complete). I've found hints that people are looking for untangling algorithms that run in polynomial time (...
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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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