Questions tagged [knot-theory]

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.

Filter by
Sorted by
Tagged with
4 votes
2 answers
333 views

Chirality and Anti-Chirality of links in 3 and in 5 dimensions

We know there is a chiral knot which is a knot that is not equivalent to its mirror image. It is well known in the mathematical field of knot theory: https://en.wikipedia.org/wiki/Chiral_knot My ...
wonderich's user avatar
  • 10.3k
5 votes
2 answers
399 views

Zero surgery on a Seifert fiber space

I have a problem with understanding what is a neighbourhood of a singular fiber in a Seifert fibered space coming from the zero surgery. For me a 3-manifold $Y$ is a SFS if it has a decomposition into ...
Stephen Dedalus's user avatar
1 vote
1 answer
239 views

Motivic knot embedding

I've been trying to be more diligent about reading motivic homotopy theory, and have been reading Levine's 'An Overview of Motivic Homotopy Theory.' I think the subject is fascinating, and I've ...
Matt Cuffaro's user avatar
1 vote
0 answers
73 views

Classification of surface curves? [duplicate]

Having a genus $g$ closed orientable surface $\Sigma$ (connected sum of $g$ tori), how do I encode any embedded closed curve up to planar isotopy? For $g=1$ we can just state the coefficient $k \in \...
Jake B.'s user avatar
  • 1,423
3 votes
0 answers
117 views

Dimension of the skein module of a closed manifold?

I'm looking for a reference to Witten's conjecture that the free part of the (Kauffman bracket) skein module of a closed 3-manifold is finitely generated, i.e. the dimension of $K(M)$, where $M$ is a ...
Jake B.'s user avatar
  • 1,423
5 votes
5 answers
2k views

Connection Between Knot Theory and Number Theory

Is there any connection between knot theory and number theory in any aspects? Does anybody know any book that is about knot theory and number theory?
Peg Leg Jonathan's user avatar
10 votes
1 answer
315 views

Is the quantum $\mathfrak{sl}_3$ invariant stronger than the quantum $\mathfrak{sl}_2$ invariant?

Both the $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$ quantum framed link invariants can be computed using linear skeins. The first being computed using the Kauffman bracket and the second using a similar ...
user530316's user avatar
1 vote
1 answer
256 views

Signature/nullity function for a link obtained by parallel pushoffs of a knot?

Let $K$ be an oriented knot in $S^3$ together with a framing $n$. Let $K(a,b)$ be the oriented link obtained by taking $a$ copies of the $n$-pushoff of $K$ with the same same orientation as $K$ and $...
user101010's user avatar
  • 5,319
9 votes
2 answers
397 views

About the commutativity of the $1^\text{st}$ homotopy group of the space of knots

I would like to know if the fundamental group of the connected component of a knot space could be non commutative. I am specially interested in the case of $\mathbb{R}^3$, $\mathbb{S^3}$ or some other ...
X1921's user avatar
  • 325
6 votes
0 answers
115 views

Is there a notion of tunnel number for 2-knots?

Given an embedded circle $K$ in $S^3$, the tunnel number of $K$ is the minimum number of embedded arcs one needs to add to $K$ so that the complement of $K$ and the arcs is a handlebody. For an ...
Román Aranda's user avatar
7 votes
0 answers
265 views

Chromatic polynomial and the circle

In https://arxiv.org/pdf/1208.5781.pdf It is proved that there is spectral sequence converging to $H^*(M^G,R)$ with the E1 page given by the graph cohomology complex $C_A(G)$ where $A:=H^*(M,R)$. My ...
Matthew Levy's user avatar
3 votes
0 answers
53 views

Subalgebras and ideals of algebra of Vassiliev invariants

Let $\mathcal{K}$ be the commutative monoid whose elements are (isotopy) equivalence classes of knots with composition under the connected knot sum, and $\mathbb{Z}\mathcal{K}$ be the corresponding ...
Meths's user avatar
  • 287
12 votes
3 answers
1k views

SL(2, C)-representation of a knot

When studying knot theory I often encounter $SL(2, \mathbb{C})$-representation of knots (of the knot group) or the $SL(2, \mathbb{C})$ character variety of a knot group. But I just don't seem to ...
Jake B.'s user avatar
  • 1,423
10 votes
1 answer
417 views

A question about dimension of SL(2,C) character variety of knot group

It is known that if there isn't a closed essential surface in $S^3 \setminus K$, the dimension of $SL(2,\mathbb C)$ character variety is $1$. (In fact, it holds for a general 3-manifold, not only for ...
Seonhwa  Kim's user avatar
3 votes
0 answers
87 views

Handlesliding a two component, linking number 1 link

Let $L = K_1 \cup K_2 \subset S^3$ be a two component framed link with $lk(K_1,K_2) = 1$. Let $\hat{L}$ denote the set of all links obtained by handlesliding $L$ around an an arbitrary number of ...
user101010's user avatar
  • 5,319
0 votes
1 answer
228 views

Proving knot polynomial dependencies and skein relations

I have two questions: From the definition of the Jones polynomial as the normalization of the Kauffman bracket $(-A^3)^{-w(D)} \langle D\rangle$ and substituting $A\rightarrow t^{-1/4}$, how does one ...
Jake B.'s user avatar
  • 1,423
11 votes
2 answers
436 views

Is there a known invariant for knotted surfaces defined by skein relations?

Is there a known invariant for knotted surfaces in $\mathbb{R}^4$ (possibly with additional structures, e.g. colored, framed, etc.) which can be defined using skein relations? By skein relations for ...
Henry's user avatar
  • 1,410
8 votes
1 answer
250 views

Set of Jones polynomials as the knot varies

Is a characterization known for the set of Laurent polynomials arising as the Jones polynomial of some knot? More generally, is such a characterization known for any of the famous knot polynomials?
pre-kidney's user avatar
  • 1,289
3 votes
1 answer
173 views

"Reidemeister moves" for self-intersecting loops

Reidemeister showed that just three basic moves (which manipulated crossings) were needed to test whether two knots were equivalent or "ambient isotopic." Suppose we expand the class of topological ...
David G. Stork's user avatar
9 votes
1 answer
359 views

Does the union of all finite groups yield a complete knot invariant for prime knots?

It is established in Whitten - Knot complements and groups together with the Gordon-Luecke theorem (that knot complements determine knot type) that the type of a prime knot is determined by the ...
Simon1729's user avatar
  • 133
10 votes
1 answer
514 views

Relation between the Casson-Gordon invariants $\sigma(M, \chi)$ and $\sigma_r(M, \chi)$

Setting: There are two objects in knot theory that are commonly referred to as the Casson-Gordon invariants: the invariant $\sigma$, and the invariant $\tau$ (see for example A. Conway’s notes ...
user331406's user avatar
22 votes
1 answer
604 views

Presenting 3-manifolds by planar graphs

From a planar graph $\Gamma$, equipped with an integer-valued weight function $d:E(\Gamma) \sqcup V(\Gamma) \to \mathbb{Z}$, one can build a $3$-manifold $M_{\Gamma}$ as follows. For each vertex $v$, ...
Lucas Culler's user avatar
1 vote
0 answers
87 views

Monoidal structure on left dg-modules over a brace algebra

Relating to my other question: Modules over Hopf Algebras and $E_2$-algebras Preliminary: Let $A$ be an associative dg-algebra that is also an algebra over the brace operad. Let $M$ and $N$ be left ...
Matthew Levy's user avatar
4 votes
1 answer
261 views

A combinatorial gadget for smoothly slice knots

As far as I searched, I couldn't find something valuable but is there any combinatorial (or computable) gadget for knots to guarantee them to be smoothly slice? For example, by the virtue of the ...
Oğuz Şavk's user avatar
  • 1,292
7 votes
2 answers
402 views

Is every element of $Mod(S_{g,1})$ a composition of right handed Dehn twists?

Let $S_{g,1}$ be the surface of genus $g \geq 1$ and $1$ boundary component. Let $Mod(S_{g,1})$ be the mapping class group in which we allow isotopies to rotate the action on the boundary (...
Paul's user avatar
  • 1,379
2 votes
1 answer
182 views

effectively distinguishing knots

It was proven, I think by Mijatović EDIT: by Waldhausen, that there is an effective algorithm for distinguishing knot complements (the effective constants were found by Coward and Lackenby). The bound,...
Dmitry Vaintrob's user avatar
2 votes
0 answers
83 views

Is there a knot invariant robust to hiding one part of the diagram behind another?

Is there a well-known knot invariant that can be computed solely by inspecting an arbitrary projection of the knot into the plane (with a marking of each crossing as "over" or "under")? The reason ...
GMB's user avatar
  • 1,379
9 votes
1 answer
660 views

Are there knots that can be distinguished by the Alexander-Conway polynomial, but not the Alexander polynomial?

On page 9 of Kauffman's Formal Knot theory, Kauffman claims The Alexander-Conway Polynomial is a true refinement of the Alexander Polynomial. Because it is defined absolutely (rather than up to ...
Kai Nakamura's user avatar
8 votes
1 answer
734 views

Modules over Hopf Algebras and $E_2$-algebras

Justin Young has a paper on the brace bar-cobar duality between hopf algebras and $E_2$-algebras: https://arxiv.org/pdf/1309.2820.pdf I was wondering if anybody knows of a nice relationship between ...
Matthew Levy's user avatar
6 votes
1 answer
235 views

Do an unlinked trefoil and figure-eight cobound an annulus in $B^4$?

Let $K_1$ the trefoil (left or right hopefully does not matter?) and let $K_2$ be the figure-eight knot in $S^3 = \partial B^4$. Are there any smooth properly embedded annulus $A$ in $B^4$ with $\...
user101010's user avatar
  • 5,319
2 votes
0 answers
90 views

Is there a measure of the failure of the Alexander polynomial to distinguish knots?

Has there been any research into something like the ratio of distinct Alexander-indistinguishable knots to total knots (up to some measure of complexity)? This was a random question asked of me by a ...
Josh Barnard's user avatar
14 votes
2 answers
729 views

Which knots are singularities of a hyperbolic cone-manifold structures on $S^3$?

Which knots $K\subseteq S^3$ are such that there is a hyperbolic cone-manifold structure on $S^3$ that has (exactly) $K$ as the singular locus? What if in Question 1 we restrict the cone angles to be $...
Giulio Belletti's user avatar
1 vote
0 answers
67 views

Common invariants of virtual knots?

When I google invariants of virtual knots (links), there's a bunch of polynomial (and other) invariants, but it's very hard to distinguish which of these are considered as somewhat classical. In the ...
Jake B.'s user avatar
  • 1,423
3 votes
1 answer
203 views

Basis for Annular Skein Algebra

Background/Notation: Given the Iwahori-Hecke algebra $H_{n}$ (over some ring commutative ring $R$ with identity) with generators $\{T_{1},\ldots T_{n-1}\}$, we know it has basis $\{T_{w}\}_{w\in S_{...
Sachin Valera's user avatar
7 votes
0 answers
206 views

IH-moves on trivalent graphs, and a complex that might be known to low-dimensional topologists

Here is a combinatorial problem which is hard to Google but seems like it might have a solution well known to people who study finite type invariants etc. Let $G_{g,b}$ denote the set of finite ...
Dan Petersen's user avatar
  • 39.2k
6 votes
1 answer
846 views

HOMFLYPT vs. Jones vs. Alexander polynomial?

I'm searching for examples (perhaps the simplest one?) to show that the HOMFLYPT polynomial is stronger than the Jones and Alexander polynomial, respectively. Any ideas what is the 1st knot in the ...
Jake B.'s user avatar
  • 1,423
6 votes
0 answers
153 views

Relation between different versions of Bar-Natan homology

In Bar-Natan's paper: Khovanov’s homology for tangles and cobordisms, he defined a deformation of Khovanov homology. Namely, for any $m\geq 0$, Bar-Natan's homology $BN^{m}(K)$ is obtained by ...
user44651's user avatar
  • 1,049
4 votes
1 answer
260 views

Framing dependence of HOMFLY polynomial

I want to understand the framing dependence of the Khovanov-Rozansky homology, and as its first step, I am trying to understand the framing dependence of the HOMFLY polynomial (i.e. quantum $sl(n)$ ...
Henry's user avatar
  • 1,410
4 votes
1 answer
229 views

Quandle homomorphism does not always induces group homomorphism on inner automorphism groups of quandles

Let $X$ and $Y$ be two quandles and $f: X \rightarrow Y$ be a quandle homomorphism. Then we can define a map $\bar f: Inn(X) \rightarrow Inn(Y)$ as $\bar f(S_a)=S_{f(a)}$, where $a \in X$. Then $\bar ...
eyp's user avatar
  • 163
4 votes
0 answers
150 views

Blow-up and Blow-down kirby local moves for non-orientable $3$-manifold

Can anyone explain or give a reference about the Blow-up and Blow-down Kirby local moves for non-orientable $3$-manifolds? Thanks, advance.
Selvakumar A's user avatar
4 votes
1 answer
566 views

Multivariate Alexander polynomial vs single variable (Conway) Alexander polynomial

I consider the multivariate Alexander polynomial $\Delta(t_1,\ldots,t_n)$ for a $n$-component link (defined using e.g. the Fox derivative). If we wish to construct a 1-variable polynomial $A(t)$, we ...
Jake B.'s user avatar
  • 1,423
2 votes
1 answer
129 views

How are characterstic polynomials (resp. Alexander polynomials) distributed amongst adjacency matrices (resp. grid diagrams)?

Fix $n$, and consider the characteristic polynomials for all $C=2^{\frac{n(n-1)}{2}}$ adjacency matrices representing undirected, unweighted graphs on $n$ vertices. Are the characteristic ...
Mark S's user avatar
  • 2,143
21 votes
2 answers
871 views

Define the 3d Chern-Simons TQFT on a discrete simplicial complex

Question: What is the challenge and the current status to define the 3d Chern-Simons(-Witten) (CSW) theory on a simplicial complex or on a discrete lattice? (Or is there a no-go or an obstruction ...
wonderich's user avatar
  • 10.3k
10 votes
2 answers
537 views

Is the Lisca-Matic bound (aka slice-Bennequin bound) strictly stronger than the Bennequin bound?

The Bennequin bound [1] says that, for a transverse knot (or later link) $K$ in $S^3$, $$\mathrm{sl}(K) \le - \chi(\Sigma)$$ for any Seifert surface $\Sigma$ for $K$, where $\mathrm{sl}$ is the self-...
Dylan Thurston's user avatar
1 vote
1 answer
119 views

"Flat links", a reference request

A hyperbolic link is one whose complement admits a hyperbolic metric. Hyperbolic links, and especially hyperbolic knots, are quite popular these days. However, I am currently interested ...
Alex Gavrilov's user avatar
2 votes
3 answers
588 views

Multivariable vs single variable Alexander polynomial for links?

If we take a $n$-component link $L$, we have the multivariable Alexander polynomial $\Delta(L)(t_1,\ldots,t_n)$. Is there a standard single-variable Alexander polynomial? If yes, is it just euqal to $\...
Jake B.'s user avatar
  • 1,423
34 votes
3 answers
4k views

Why should I care about the Jones polynomial?

The invention of the Jones polynomial led to hundreds of papers and a Fields medal. However, as far as I can tell it had few consequences in topology. After all, after Thurston’s work we already ...
Julia's user avatar
  • 387
2 votes
0 answers
207 views

What is an ambient isotopy categorically?

Let $\mathcal T$ be a category of "nice" topological spaces (CW?) and continuous maps between them. We can construct the homotopy category $\mathrm{Ho}\mathcal{T}$ which gives for any two objects $X,Y$...
CobarOfSoap's user avatar
7 votes
1 answer
312 views

What is the complexity of determining if a knot group is $\mathbb{Z}$?

It is known from the work of Waldhausen that the isomorphism problem for knot groups is decidable. What is then: The complexity of determining if a knot group is $\mathbb{Z}$? .i.e. same as the ...
Prathamesh's user avatar
9 votes
3 answers
300 views

Knots of fixed genus with arbitrarily large volume

Consider all knots with fixed genus $g\ge 2$ (I am considering the classical 3-genus). Do there exist infinite families of genus $g$ knots with arbitrarily large volume? The answer seems like it ...
Rocket Man's user avatar

1
5 6
7
8 9
19