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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Does every knot contain all four vertices of an isosceles trapezoid?

I ask this question with some trepidation, because it may be trivial and/or of entirely recreational interest. Erika Pannwitz proved in 1933 that every non-trivial knot contains a quadrisecant (four ...
3
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0answers
334 views

Positivity of braid monodromy of curve singularities

I recall the notion of braid monodromy. Let $C \subset \mathbb{C}^2$ be an algebraic curve, and choose a projection $\pi: \mathbb{C}^2 \to \mathbb{C}$ such that the restriction $\pi: C \to ...
3
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2answers
365 views

Construction of Kirby-diagram for slice-complement

Let $(S^3,K)=\partial (D^4,D)$ be a slice knot. (i.e. a knot in $S^3$ bounds a disk in $D^4$.) Are there any relation between the knot diagram of $K$ and the kirby diagram for $(D^4-D,M_K)$, where ...
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1answer
813 views

smooth proof of reidemeister theorem

Hello! In another post Ryan Budney has mentioned a "smooth proof" of the theorem of Reidemeister... (@Ryan Budney:) Do You know if there is a book (or paper) containing such a proof? I am interrested ...
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3answers
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Is anything known about this braid group quotient?

Let $B_n$ be the braid group on $n$ strands. As is well known, if $\sigma_i$ is the operation of crossing the string in position $i$ over the string in position $i+1$, then the elements ...
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1answer
230 views

“Skein” equations sets that can reduce any graph

Consider for example the approach to the Kuperberg G2 or the Yamada polynomial. I don't know whether relations of graph (in contrast to knot) theory are also usually called "skein" but I simply carry ...
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2answers
420 views

Which knots can appear as a space-time cut/slice of an open trefoil?

An open trefoil is a trefoil tied in an infinitely long line. An open trefoil that is at rest in (flat) space is a surface in space-time. Now, other observers in space-time have other time ...
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1answer
260 views

Can a closed trefoil appear as a space-time “cut” of an open trefoil?

An open trefoil is a trefoil tied in an infinitely long line. An open trefoil that is at rest in (flat) space, in space-time is a knotted hypersurface. Different observers in space-time have ...
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2answers
995 views

How many lanes has a freeway? (Crossing free Kuperberg G2, that is.) EDITED

For referencing, I keep the original title and post and ask only about the simplest case (and forget the freeway with crossings for now). Consider a trivalent graph, e.g. the dodecahedron or cube ...
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1answer
382 views

What is the shape of a tight open trefoil?

Take an infinitely long rope of diameter 1, ideally flexible and ideally slippery. Tie a trefoil knot into it and pull it tight. Describe the resulting rope shape analytically. The problem is ...
31
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9answers
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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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297 views

A knot theory form of the carpenter's ruler question

Hey, The carpenter's ruler problem is about polygons you can make with planar joints. You can formulate a similar question for knots when you introduce ball joints: Given m ball joints connected by ...
2
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0answers
128 views

Simple terminology question about the Dubrovnik (Kauffman) polynomial

In my S matrix classification attempts I encounter a lot of Dubrovnik polynomials of the form D(z-1/z,z^n) and D(-z+1/z,z^n). [Second variable is for writhe, n is an integer; for the first I don't ...
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1answer
312 views

Handle slides homeomorphism

For handle decomposition of surface, suppose I have a twisted 1-handle(twisted only once) adjacent to an isolated pair of linked handles, is the handle slide operation enough to move the previous one ...
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165 views

On Birman-Wenzlyfying the B2 spider

Prelude: First of all, let "S matrix" denote "an abstract 4D tensor satisfying the usual isotypy rules (with no arrows!)". I'm busy trying to classify all possible S matrices (paper pending) - it's ...
5
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1answer
541 views

Link of singularities

For an isolated plane curve singularity, given by homogeneous equation $F=0 \subset \mathbb{C}^2$, one consider the curve $(F=0) \cap S^3 \subset S^3$, and we call it the link of singularity. some ...
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1answer
339 views

Braid*Temperley-Lieb=?

I would be very astonished if this algebra isn't named. You simply have the braid AND the Temperley-Lieb generator in the algebra. Rules are the usual Reidemeister equivalents plus the kink and ...
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2answers
766 views

Is there a table of (fibred knot) monodromies?

Background/motivation I'm working on contact topology (in dimension three): a fundamental theorem of Giroux gives us a bijection between contact structures (up to isotopy) and open books (up to ...
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1answer
782 views

Is the HOMFLY Polynomial the best knot invariant? [closed]

Is the HOMFLY Polynomial the best polynomial invariant that can be calculation from skein relation?
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283 views

Properties of the Jones polynomial

Let $V(L)$ be the Jones polynomial of the oriented link $L$. For $\alpha \in B_n$, we write $V(\alpha)$ for $V(\hat{\alpha})$, where $\hat{\alpha}$ is the closed braid associated to $\alpha$. The ...
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162 views

Polynomial knots

I am an undergraduate studying polynomial knots. Given a polynomial knot, it is common to generate a tube around the space curve for visualization purposes. I define the set T[c(t)] as the set of all ...
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223 views

“Lorentz-invariant” Morse theory possible?

Hi, I hope filing this under "Morse theory" is correct. For the "abstract tensor" approach to knot theory you need: a) a 4D tensor to emulate particle interaction, b) a 2D tensor to emulate pair ...
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2answers
417 views

Do the results of (1/n)-surgery determine the link?…

Knowing the result of knot surgery is often not enough to determine the knot. Indeed, there are 3-manifolds admitting an infinite number of descriptions as surgery on a (1-component) knot in $S^3$. ...
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2answers
382 views

Algorithm for detecting ribbon or slice links?

A link in $S^3$ is said to be slice if it bounds a collection of flat disks into the $4$-ball. Here "flat," means that there is a (locally) trivial normal bundle. This condition can be strengthened to ...
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738 views

Whitehead doubles of any knots

I was curious about the fact that the Whitehead doubles of all knots have Alexander polynomial equal to 1, which is the same as a unknot. How to prove this?
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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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2answers
405 views

Is every virtual knot group an HNN extension?

A basic fact in knot theory is that a knot group $\pi(K)$ is an HNN extension of $\pi(F)$, the fundamental group of a Seifert surface complement. A nice discussion of this may be found in Chapter 11 ...
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1answer
540 views

Markov Trace and Markov Property

Hey guys, I'm a computer science student attempting to understand a quantum algorithm that uses braid theory - something I'm completely unfamiliar. I've getting through the algorithm but I can't ...
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4answers
890 views

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 ...
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1answer
554 views

monodromy defects and Chern-Simons

In the context of string theory I recently read "The formulation of Chern-Simons theory in terms of monodromy defects can be carried through all the dualities of the present paper, leading to ...
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2answers
347 views

What is known about links with a countably-infinite number of tame components?

I want to be clear about my phrasing, but I'm not a topologist, so when I say "knot" or "link" I mean the equivalence class under ambient isotopy of an embedding of the circle into $\mathbb{R}^3$. ...
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1answer
527 views

Length of shortest possible knot

Consider a line L in R^3 in the shape of a trefoil knot. Consider the surface S that is the union of all unit circles that have centers on this line and whose tangent vectors are all perpendicular to ...
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305 views

Ramified cover of 3-ball

I it true that any connected oriented 3-manifold with nonempty boundary can be obtained as a ramified cover of 3-ball with a ramification at a link? link = a 1-dimensional submanifold with possibly ...
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1answer
216 views

Simplified Jones trace invariant for links

Jones (1985) defines a simplified trace invariant for knots by $W_K(t)=\frac{1-V_K(t)}{(1-t^3)(1-t)}$. Then, e.g., the Arf invariant for $K$ is $Arf(K)=W_K(i)$. Does this work for oriented links as ...
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1answer
220 views

Are braid links proper links?

Are braid links proper links? Or are the concepts involved unrelated?
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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 ...
18
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3answers
2k views

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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5answers
880 views

Can surfaces be interestingly knotted in five-dimensional space?

It's possible this question is trivial, in which case it will be answered quickly. In any case, I realized that it's a basic question the answer to which I should know but do not. Everybody loves ...
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2answers
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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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1answer
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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 ...
3
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1answer
322 views

Homology boundary links

According the the original definition by Smythe, a homology boundary link $L\subseteq S^3$ with $m$ components is a link which sarisfies one of the following equivalent conditions: (1): The ...
4
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1answer
333 views

Cubic skein relations

Hi, please note that this question deals with undirected knots/links! The most generic cubic skein relation for a knot polynome would be where w^3 is one positive writhe unit. The form is fairly ...
3
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1answer
494 views

Boundary links and ribbon links.

This question is about the relation between the notions of boundary link and ribbon link. For the definition of ribbon link see: ribbon links - counterexamples. An n-component link ...
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278 views

Computing Quantum Dimensions

Hi, in "Jaeger’s Higman-Sims state model and the B2 spider" by Greg Kuperberg (arxiv:math9601221v1, 1996) there are some quantum dimensions listed in the "Discussion" part. Evidently quantum groups ...
5
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1answer
476 views

When is a connected sum of torus knots a slice knot?

This question is about the beaviour of 4-genus of knots with respect to connected sum. Let us indicate with $T(k)$ a Torus knot of type $(2,k)$, $k$ is an odd integer. Fix an orientation for every ...
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1answer
605 views

isotopy doesn't make sense (Milnor)

hello, I am having a hard time following this isotopy put forth by Milnor in On the Total Curvature of Knots For each $c$ and $p$ in $\mathbb{R}^{n-1}$ such that $\|c-p\| > < r$, there is ...
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1answer
181 views

crookedness of convex curves (milnor)

hello, I'm currently reading On the Total Curvature of Knots and am trying to understand one of the lemmas in it (3.3) A closed polygon $P$ in $H^2$ is convex if and only if for every $b$ either ...
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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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4answers
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Utility of virtual knot theory?

Virtual knot theory is an interesting generalization of knot theory in which ``virtual" crossings are allowed. See Kauffman's Virtual Knot Theory for an introduction. Greg Kuperberg gave a nice ...
13
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832 views

Are there moves between Reidemeister moves?

Background Knots are typically written in 2 dimensions as a loop in the plane with normal crossings. One then asks when two such diagrams describe the same knot. Two diagrams describe the same knot ...