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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How to visulize surface link in four dimension?

I am now facing a problem with "surface link" in four dimension. I have heard that three 2-torus can be linked in four dimension. And I have created a movie by cutting four dimensional space with ...
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176 views

Links which HOMFLY homology distinguish but the HOMFLY polynomial does not.

Does anyone know of a pair of different links which the HOMFLY polynomial does not distinguish, but HOMFLY homology does? Or does there exist such a pair of links? I'm assuming there does exist ...
3
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165 views

Satellite knot example

Can someone provide me with an example of a satellite knot with symmetry group which is neither cyclic nor dihedral?
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231 views

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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3answers
581 views

Visualize Fourth Homotopy Group of $S^2$

I know $\pi_4(S^2)$ is $\mathbb{Z}_2$. However, I don't know how to visualize it. For example, it is well known that $\pi_3(S^2)=\mathbb{Z}$ can be understood by Hopf Fibration. Elements in ...
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Fibered knot with periodic homological monodromy

It is well-known that there exist pseudo-Anosov automorphisms of surfaces that act trivially on the homology: they form the Torelli group. Similarly there exists pseudo-Anosov automorphisms that act ...
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Oriented Reidemeister move R2d by splices and loop adding/erasing?

Consider the move R2d which applies to tangle diagrams, described in the Figure 1. Question: is it possible to achieve the move R2d by a sequence of splices and loop adding or erasing, i.e. is ...
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274 views

Is the connected sum of knots an isometry?

Take $X$ as the set of knots in the 3-sphere (i.e. smooth embeddings of $S^1$ in $S^3$ up to smooth isotopy), endowed with the Gordian distance $d$. For a fixed knot $K$ we can define the map ...
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354 views

Dehn presentation of a knot group

The knot group is the fundamental group of the knot complement in $S^{3} $. The Dehn presentation of the knot group is a particular group presentation obtained by looking at the regions and crossings ...
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376 views

Diagrammatic proof of unique prime decomposition of knots

Consider a knot to be a diagram in a plane--- i.e. a drawing of a finite connected planar graph (loops and multiple edges allowed) whose vertices are 4-valent with cyclic ordering for the incident ...
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164 views

Locally flat non-smooth discs

There are many knots (e.g. the $P(-3,5,7)$-pretzel knot) that are topologically, but not smoothly slice; "topologically" slice means that there is a locally flat embedding of a disc into the 4-space, ...
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Genus one fibered links

It is well-known that the only genus one fibered knots are the trefoil and the figure-eight. On the other hand, there exist infinitely many fibered links for any fixed higher genus. My question is ...
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Categorification of WRT invariants of integral homology spheres

First, I would like to know how many definitions are there for categorification of WRT invariants. In addition, I wonder if the categorified version of WRT invariants have been explicitly computed for ...
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Can all knot polynomials derived from Skein relations be categorified?

Can all knot polynomials derived from Skein relations be categorified?
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159 views

Seifert genus of the lift of a knot in its cyclic branched covers.

I was wondering if there are any known examples of knots $K$ in $S^3$ with Seifert genus $g$ so that the lift of $K$ sitting inside its $n$-fold cyclic branched cover bounds an embedded surface of ...
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108 views

Are isolated knots in tangles the same as ordinary knots?

Suppose you have an arc in a ball with ends fixed on a boundary. In other words, we have an isolated knot on one of the arcs of a tangle. If we glue the ends we get an ordinary knot and clearly ...
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336 views

Khovanov-Rozansky homology and spectral sequences

In arXiv:math/0607544 (following conjectures in arXiv:math/0505662), Rasmussen constructs a family of spectral sequences (the "d_N differentials"), starting at the HOMFLY homology of a knot, and ...
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1answer
117 views

One question about the quandle

Given a finite quandle $Q$, for any knot $K$ one can associate an invariant, i.e. the number of proper colorings $p(K)$. Let us consider the inverse $K^{-1}$ and mirror image $K'$ of $K$. My queston ...
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236 views

Existence of different knots in $RP^3$ having the equivalent liftings in $S^3$

I'm looking for the answer to following question. Do exist different knots in $RP^3$ which have equivalent liftings in $S^3$ under covering $p:S^3\rightarrow RP^3$?
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171 views

Differences between various categories of surface embeddings in 4-space

This is a very naive question, but I'm trying to understand the difference between the various categories when it comes to embedding surfaces in 4-dimensional manifolds. The situation I'd really like ...
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110 views

Some questions about ideal knots

The ropelength of a knot curve $C$ is defined as the ratio $L(C) = \lambda(C)/ \tau(C)$, where $\lambda(C)$ is the length of $C$ and $\tau(C)$ is the thickness of the knot defined by $C$ [from ...
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241 views

Vassilliev invariants of knots and their cables

The following is perhaps a standard question, but i could not find a plain enough answer by simply searching online. Q: Given a knot $K$ and its $(p,q)$-cable $K_{p,q}$ what is a relation between ...
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integer surgeries on knots

I have constructed a list of surgery coefficients which yield spherical space forms. For instance, there are two knots with different Alexander polynomials on which 29-surgery will give a small ...
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85 views

Difference between $MCG(ext(K))$ and $MCG(S^{3} - K)$

Can someone explain the difference between the mapping class groups of the knot complement and knot exterior?
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322 views

Is a knotted trivalent graph determined by its set of unzips?

A knotted trivalent graph (KTG) is a framed embedding of a trivalent graph in $\mathbb{R}^3$, modulo ambient isotopy. I believe they were first considered (in the quantum topological context, at ...
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Original proof of the existence of Seifert surfaces

I read on Wikipedia that Frankl and Prontrjagin were the first to prove that a link $\mathbb{R}^3$ bounds a surface. A few years later Seifert published a proof using the "Seifert algorithm" which ...
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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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250 views

Hyperbolic Brunnian links and rectangular cusp shapes

My question is as follows. Does every hyperbolic Brunnian link have rectangular cusp shapes on all its components? Here is what I mean: The Borromean rings form a famous link $B$ (a smooth ...
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196 views

Reference for a proof of the Dehn presentation

I would like a reference for a proof that the Dehn presentation is a presentation of the fundamental group of the knot complement in $\mathbb{S}^{3} $.
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1answer
221 views

Can we get the HOMFLY polynomial for a torus knot from the Kauffman Polynomial?

This is essentially a yes/no/reference request question. I posted it on math.se and left it there for 5 days before posting here. Let me first just ask my question: Is there a known relationship ...
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2answers
164 views

A question on (1,1) bridge Knot

Hi, everyone. I am interested in the complement of (1,1) bridge knot in a lens space, $S^{3}$. Is there one (1,1) bridge knot in $S^{3}$ or lens space such that its complement is hyperbolic? ...
4
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1answer
195 views

Equivalent to Oriented knot complement conjecture

I would like to see why the following two statements in Kirby's list of problem are equivalent: Statement 1: If $K_1$ and $K_2$ are knots in a closed oriented 3-manifold $M$ whose complements are ...
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224 views

A Degree of an Arbitrary Polynomial Knot

Here a degree of a polynomial knot is a minimal degree which can define a long knot. I would like to find out how this degree can be bounded below, according to the number of crossing points, for ...
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401 views

Links with same Jones polynomial

Is there anything known about when two links have the same Jones polynomial (beyond a calculated list of small actual examples)? The first thing I would try is to compute the (formal - you would have ...
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1answer
230 views

How many 3-flip Mobius strip knots are there?

Take a clock-wise 3-flip mobius strip, Cut it down the middle and then let the ribbon cross itself 6 times. This forms a framed knot of which there are many. Get the knot diagram. I've found ...
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Things you can do with the self-writhe

I hope "self-writhe" is the established word. (0 for link-crossing, otherwise identical to writhe +1 or -1) I bet the following is known: Take some crossing of a link with self-writhe $w_a$. Flip it ...
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What is knot contact homology?

Recently, it was conjectured by the paper of Aganagic and Vafa that the $Q$-deformed $A$-polynomials can be identified with the augmentation polynomials of the knot contact homology. The $Q$-deformed ...
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1answer
176 views

Thurston-Bennequin number vs. checkerboard coloring difference

For an alternating knot K, checkerboard-color the knot (if this is a lousy ASCII crossing: %, white goes to the left/right and black to top/bottom). Assume no surplus Reidemeister 1 kinks exist (K has ...
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1answer
305 views

Problems about the Estimate the Unknotting Number

For the definition of unknotting Number, you can assess http://www.popmath.org.uk/exhib/pagesexhib/unknum.html My question is: For given a knot K, let n be the crossing number of K, is their any ...
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What is the historical connection between Zeeman's twist spinning and Fox's Examples?

Both Ralph Fox and (at that time, yet to be knighted) Sir Christopher Zeeman attended the 1961 Georgia topology conference. Fox's paper from that conference was his seminal work, "A Quick Trip through ...
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Generalizing the Reshitikhin-Turaev construction possible?

OK, I have to ask a dumb question again: Where do Lie groups enter in the construction of the Reshitikhin-Turaev invariant? The parts of the proof I understand are that 6j symbols take care of ...
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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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1answer
323 views

Can Reidemeister 3 be weakened?

If you take the diagram of the Reidemeister 3 move and "shortcircuit" two ends, you get (click http://imgur.com/kRvZa if Imgur hotlink doesn't work): ...
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How much do homological knot invariants improve the classification problem of knots?

The mutation operation in knots appears to be detected by the Floer homological invariants. See the papers by Ozsvath, Szabo and by Baldwin, Gillam. In addition, the Khovanov homology turns out to be ...
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1answer
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$6j$-symbols for $U_q({\mathfrak{sl}}_n)$ and colored HOMFLY polynomials

Explicit expression of quantum $6j$-symbolos for $U_q({\mathfrak{sl}_2})$ have been known due to the work of Kirillov and Reshitikhin. My Question: How much are known about quantum $6j$-symbolos ...
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1answer
171 views

Application of a quandle cocycle invariant for virtual knots

In knot theory, a quandle cocycle invariant was defined. Moreover, to virtual knot theory it was generalized by avoiding for virtual crossings. Question Are there many application of a quandle ...
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317 views

Infinite knot composed of parallel helices

I am wondering if there is a developed theory of "infinite knots" that could capture this object, and tell me something of its knot properties. Imagine vertical helices in $\mathbb{R}^3$, each ...
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432 views

Circle-arc number of a knot

I would like to build knots in $\mathbb{R}^3$ from arcs of unit-radius (planar) circles, joined together at points where the tangents match. Thus the knot will have curvature $1$ at all but the ...
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1answer
206 views

Validity of generalized Reidemeister moves for a virtual knot

I am studying virtual knot theory. A virtual knot is a knot diagram with real or virtual crossing information. The equivalence relation includes generalized Reidemeister moves. There are premitted ...
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Knot diagrams, sets of moves and equivalence relations

Short version: Does anyone study equivalence classes generated by a given set of "moves" (in the sense of, but not limited to, Reidemeister moves) on the set of knot diagrams? Yes, I understand that ...