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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What is the complex structure on the boundary torus of a hyperbolic knot complement?

Let $K$ be a hyperbolic knot in $\mathbb S^3$. Restrict the corresponding representation $\pi_1(\mathbb S^3\setminus K)\to\operatorname{PSL}(2,\mathbb C)$ to the fundamental group of the boundary ...
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Is the category of tangles that includes, X, Y, and Lambda a free Frobenius braided category?

Consider the category whose objects are non-negative integers that are represented as dots along a line, and whose morphisms are generated by $X$---positive crossing, $\bar{X}$ --- negative crossing, ...
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489 views

Is the complete functorial structure for Khovanov--Lee homology known?

I'm interested in Lee's modification of Khovanov homology, which I'll denote $\operatorname{Kh}_{\operatorname{Lee}}^\ast$. Below $L$ is a link in $\mathbb R^3$. The groups ...
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subset embedding gives trefoil knot [closed]

Let $X$ be a topological space and $E_n(X)$ the space of finite sets of cardinality $\leq n$. It is a theorem of Bott that $E_3(S^1)=S^3$. What is the idea to show that the embedding ...
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928 views

What is the three-dimensional hyperbolic volume of a four-manifold?

Every smooth closed orientable 4-manifold may be constructed via a handle decomposition. Before asking a couple of questions, I recall some well-known facts about handle-decompositions of 4-manifolds. ...
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445 views

Knot Numerology

EDIT: Ok, I condense it to only that what is needed. Assume it's possible to use the method described here Matrix decomposition the other way to decompose a $S$ matrix from knot theory. Then each ...
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530 views

Trace identities and the Kauffman Bracket skein module

Let's consider $K_t(M)$, the Kauffman bracket skein module (see this and this papers) of a three-manifold $M$. When $t=-1$, $K_t(M)$ is easily seen to be isomorphic to the ring of functions on the ...
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1answer
182 views

Invariants of unframed, oriented links from Reshetikhin Turaev construction

Hello! I have a few questions on Reshetikhin Turaev invariants. By RT any ribbon category ${\mathcal C}$ yields an invariant of oriented, framed links labelled with objects of ${\mathcal C}$. Is ...
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621 views

Are there any very hard unlinks?

This question is closely related to a question of Gowers: Are there any very hard unknots? . I'm thinking about how to create interesting knots from small numbers of local moves on unlinks. The ...
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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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A tangle matrix

The image should already say everything. But in case it is mute... :-) List tangles with 2n legs downward as column header, and with 2n legs upward as row header. (n=3 here.) In the crossing point ...
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is twisted torus knot T(7,2;4,1) prime?

what kind of twisted torus knot is prime? even more , is twisted torus knot T(7,2;4,1) prime?
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303 views

A question about Dehn filling in the unknot.

Hello, If you have a link of 2 components $U$ and $V$ in $S^3$. $U$ is the unknot, and you make $1/q$ dehn filling in $U$, you can visualize the resulting knot $V'$ from $V$, by twisting it $q$ ...
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1answer
445 views

Now that I got a mutant-discriminating invariant…

...what can I do with the darn thing? Background: I read that still no Vassiliev Invariant with mutant-discriminating power is known (correct me if this is outdated). Now, my research lead to a whole ...
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1answer
322 views

Some Vassiliev Invariant Questions

The V.I. definition goes doublepoint=overpass-underpass (or was it the other way around? If it's 50:50, I score 0 always :-). Would it lead anywhere to define doublepoint=overpass+underpass? (Even if ...
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550 views

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 ...
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339 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
372 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 ...
5
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831 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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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
426 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
263 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
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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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383 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 ...
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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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302 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 ...
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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
320 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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166 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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572 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
342 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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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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812 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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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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227 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
419 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
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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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3answers
759 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?
26
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3answers
979 views

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 ...
8
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2answers
415 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
579 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
905 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
558 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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356 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 $\mathbb{R}^3$. ...
6
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1answer
529 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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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
218 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
221 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 ...