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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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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1answer
177 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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0answers
114 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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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
147 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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1answer
238 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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1answer
201 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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1answer
122 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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2answers
292 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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1answer
320 views

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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348 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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2answers
336 views

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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4answers
827 views

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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2answers
285 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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1answer
201 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
245 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
177 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? ...
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1answer
217 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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1answer
229 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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3answers
470 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
271 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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157 views

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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1answer
239 views

Unknotting number and crossing number

It is well known that if $c(K)=2n+1$, then $u(K)$ is less than $n+1$. It can not be sharper because of the trefoil knot. On the other hand, if $c(K)=2n$, then similarly we have $u(K)$ is less than ...
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1answer
630 views

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
197 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
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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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1answer
152 views

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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2answers
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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
334 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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1answer
485 views

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
176 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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2answers
335 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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3answers
455 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
225 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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4answers
531 views

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 ...
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796 views

Knot theory without planar diagrams?

I remember reading somewhere that there are just a few contributions to knot theory which do not involve knot diagrams. Hence my question: Does anybody know about papers concerning knot theory which ...
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278 views

On finding A-polynomials

I have two questions to obtain the explicit forms of A-polynomials. Takata used the mathematica pacage qMultisum.m to obtain the recursion relation of the colored Jones polynomials for twist knots. ...
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488 views

Categorification of finite type invariants

Recall that finite type invariants are those numerical knots invariants vanishing on knots with enough singularities. The value of an invariant on a singular knots is computed using the Vassiliev ...
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3answers
444 views

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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1answer
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Computations and applications of Khovanov's functor valued invariant of tangles

Soon after his famous paper "A categorification of the Jones polynomial", Khovanov introduced a "bordered" version. His theory assingns to every oriented even tangle a complex of (H_n,H_m)-bimodules, ...
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1answer
253 views

On expressions of colored Jones polynomials

In the paper by Masbaum, it was shown that the colored Jones polynomials for a twist knot $K_p$ can be written as \begin{eqnarray} J_{n}(K_p;q)&=&\sum_{k=0}^{\infty} {\cal C}_{K_p}(k) ...
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1answer
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Euclidean symmetries of torus links in R^3

I have a question about whether Ryan Budney's question: Torus knots in Euclidean space -- a symmetry argument can be extended to links. He asks: Suppose you have a $(p,q)$ torus knot $K$ in ...
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1answer
659 views

Homology and homotopy type for knot complements

I'm trying to understand a paper by Kawauchi and Matumoto, 'An estimate of infinite cyclic coverings and knot theory.' In one of their proofs they have a manifold $E$ which is the complement of a ...
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331 views

High-dimensional ribbon knots

Let us suppose that we have a ribbon embedding $S^n \rightarrow S^{n+2}$ for $n\geq 3$. Call this knot $K$. By a theorem of Levine (and Trotter for $n=3$ I believe) we know $K$ is unknotted if the ...
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1answer
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Resemblance between Birman-Murakami-Wenzl algebra representations and the Lawrence-Krammer representations

At the end of Stephen Bigelow's paper "Braid Groups are Linear", he mentioned that there is a striking resemblance between the matrices of the Lawrence-Krammer (LK) representations and "those of a ...
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Minimal piecewise-linear knot diagram [closed]

I'm looking for an answer to the following question: Given a knot in $\mathbb{R^{3}}$ can we find a piecewise-linear diagram of it wich is minimal (has a minimal number of verticies)?