4
votes
1answer
118 views

Do we get a instanton $S^{3}$ if we do $1/n$ surgery on a knot in $S^{3}$?

Consider the following question: If $K\subset S^{3}$ is a nontrivial knot. Let $Y$ be the manifold obtained by doing $1/n$-surgery ($n\geq1$). Is it possible that the instanton Floer homology of $Y$ ...
1
vote
1answer
99 views

Clarification of Gabai's exposition of Murasugi Sums in 'the Murasugi sum is a natural geometric operation'

Gabai states that the Murasugi sum of two hopf bands yields a spanning surface of either the figure eight knot, the trefoil knot or a link of three components. Figure one shows two oppositely twisted ...
6
votes
0answers
107 views

Does the shortest path between two braids pass through string links?

One of the fundamental facts underlying the application of braid theory to knot theory is that braids inject into string links. This means that braids $B_1$ and $B_2$, considered inside a cube $I^3$, ...
7
votes
1answer
411 views

Is this knot invariant already treated somewhere in the literature?

Fix a knot type $K \subset S^3$, and consider the set $$Y_K = \{ \mbox{Diagrams of }K \} / \mbox{planar isotopy}.$$ We can turn $Y_K$ into a metric space by considering the distance induced by ...
3
votes
2answers
137 views

Handlebody decomposition of a 3-manifold adapted to a link

Given a compact connected 3-manifold $M$ with non-empty boundary, and a link $L \subset M$, is there a handlebody decomposition of $M = H^0 \cup (\cup_i H^1_i) \cup \{\text{2-handles}\}$ such that: ...
4
votes
0answers
69 views

Ribbon knot presentations

Suppose $K$ is a $n$-knot, $n\geq 2$, which bounds two different ribbon disks $D_1, D_2$. These ribbon disks induce unique ribbon $(n+1)$-knots $K_1, K_2$ respectively. Is it known whether $K_1$ and ...
1
vote
0answers
80 views

Entangled helical knots

Consider a pair of disjoint, congruent helices $H_1$ and $H_2$ passing through one another in the following sense. (Caveat lector: This question is not of general interest! It is also long.) $H_1$ is ...
8
votes
3answers
284 views

What constant ensures hyperbolicity of Dehn surgery?

I am interested in showing that certain knots having a surgery description are hyperbolic. Unfortunately I have not had time yet to read Thurston's work, so my understanding of this is vague. But from ...
4
votes
1answer
148 views

The cyclic branched covers of “simple” knots in $S^3$

Is there a convenient place in the literature where the geometric decompositions of cyclic branched covers of $S^3$ branched over "small" knots is recorded? By small knots, I'm referring to things ...
7
votes
1answer
157 views

Is there a combinatorial version of PL ambient isotopy in dimension $>3$?

The classical PL Reidemeister Theorem reads: Reidemeister Theorem: Two knots in $S^3$ are PL ambient isotopic if and only if any diagram of one can be transformed into a diagram of the other by ...
4
votes
2answers
144 views

Getting surgery link from Heegaard splitting

From Lickorish-Wallace theorem, every 3-manifold is an integral surgery on a link in $S^3$. From its proof from Saveliev's book, it seems obvious that if I know the Heegaard splitting of a closed ...
7
votes
2answers
203 views

How do I efficiently find a sequence of Reidemeister moves between equivalent link diagrams?

In knot theory, two link diagrams are equivalent if and only if they can be related by performing a finite number of Reidemeister moves. But sometimes it is so confusing that I don't know which type ...
16
votes
3answers
515 views

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, ...
3
votes
0answers
107 views

More questions about high-dimensional knot invariants

In a question yesterday I asked about the existence of algebraic invariants for embeddings of n-manifolds into n+2-spheres. The answers in the positive dimension all made certain assumptions about ...
9
votes
3answers
288 views

Invariants of high-dimensional knots

In the study of knots in three dimensions, it can be shown that the fundamental group together with a specification of a meridian and longitude form a complete invariant for knots. What is known about ...
1
vote
2answers
273 views

Commutativity in the Fundamental Group and Knot Theory

Let $M$ be a connected $3$-manifold and let $\alpha$ and $\beta$ be elements in $\pi_1(M)$. Then $\alpha$ and $\beta$ can be represented by two knots $a$ and $b$ in $M$. We may further require that ...
16
votes
3answers
370 views

Looking for “large knot” examples

This question is about knots and links in the 3-sphere. I want to find an example of a "large" knot or link with some special properties. I'm looking for some fairly specific examples, but I'm also ...
4
votes
0answers
98 views

Is the normalised Kauffman bracket more powerful than the Kauffman bracket?

The Kauffman bracket polynomial for a knot diagram $D$ is a Laurent polynomial $\langle D \rangle \in \mathbb{Z}[A, A^{-1}]$. Although it is invariant under Reidemeister moves of type II and III, ...
7
votes
2answers
162 views

How does Thurston's Orbifold Geometrization imply that knots with meridional rank 2 are 2-bridge?

Problem 1.11 of Kirby's list asks whether every knot that has a knot group which can be generated by n meridians, but not less than n, is an n-bridge knot. There is a one-sentence update, saying that ...
1
vote
1answer
128 views

Two links with the same signatures but unknown if they are related by Kirby moves

I am wondering if there are links $L_1, L_2$ in the sphere $S^3$ such that: the signatures of $L_1, L_2$ are known. we do not know if they are related by Kirby moves. If so, could you specify the ...
3
votes
0answers
109 views

Is a generic link diagram semi-adequate?

Each crossing in a link diagram of a link $L$ has an A-resolution and a B-resolution. Resolving all crossings gives a collection of circles in the plane, connected by dotted lines. A state of a ...
5
votes
1answer
154 views

Reference for a theorem on crossing changes of links

I've recently stumbled upon a paper of Scharlemann on crossing changes: link text In particular I am interested in understanding Theorem 2.2 (page 6): "Theorem: If links A and B are related by a ...
12
votes
2answers
311 views

Semidirect product decomposition of the Borromean rings group

Let $X=S^3\setminus B$ be the link complement of the Borromean rings. Then $G=\pi_1(X)$ has a presentation of the form $$ G = \langle \; a,b,c \mid [a,[b^{-1},c]],\; [b,[c^{-1},a]], \; ...
2
votes
1answer
118 views

Is every quasipositive knot strongly quasipositive?

A link is called quasipositive if it has a special braid diagram, namely a product of conjugates of the positive standard generators of the braid group. If this product only contains words of the form ...
8
votes
3answers
385 views

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 ...
9
votes
0answers
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 ...
7
votes
1answer
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 ...
5
votes
1answer
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, ...
10
votes
2answers
246 views

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 ...
4
votes
1answer
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 ...
8
votes
2answers
334 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 ...
5
votes
1answer
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 ...
4
votes
2answers
240 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 ...
8
votes
3answers
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 ...
7
votes
2answers
303 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 ...
1
vote
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? ...
3
votes
1answer
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 ...
5
votes
3answers
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 ...
0
votes
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 ...
1
vote
0answers
152 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 ...
4
votes
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 ...
7
votes
0answers
212 views

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 ...
33
votes
2answers
1k views

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 ...
2
votes
1answer
170 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 ...
0
votes
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 ...
7
votes
4answers
502 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 ...
8
votes
0answers
438 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 ...
10
votes
3answers
384 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 ...
5
votes
1answer
266 views

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, ...
5
votes
1answer
307 views

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 ...