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.
920
questions
0
votes
0
answers
104
views
Name for homotopy totalization of Goodwillie tower (in embedding calculus)
Let $M,N$ be a manifold and consider the presheaf of spaces $\textrm{Emb}(-, N)$ on the open sets of $M$. Classical embedding calculus produces a goodwillie tower
$$ \ldots \rightarrow T_{k+1} \textrm{...
- 1,944
9
votes
2
answers
552
views
Bing sling isotopy to unknot
Rolfsen asked the question as to whether any knot is topologically isotopic to the unknot. Where a topological isotopy is a continuous path in $\operatorname{Emb}(S^1,\mathbb{R}^3)$.
From now on I ...
- 333
3
votes
1
answer
208
views
Picturing twisting of strands explicitly after blow downs
In order to simplify Kirby calculus proofs, one can use a box notation which indicates a number of full twists up to sign. In the scenario of two strands with twist boxes, it is straightforward to ...
- 26.4k
9
votes
1
answer
223
views
Links and non-orientable surfaces
Let $\Sigma \subset \mathbb{R}^3$ be a compact embedded surface with boundary $\partial \Sigma$ and $i:\Sigma\setminus \partial\Sigma \to \mathbb{R}^3 \setminus \partial\Sigma$ the inclusion.
Is the ...
- 26.4k
4
votes
1
answer
169
views
Alexander polynomials for a certain family of closed braids
Let $n\geq 3$ be a positive integer and $\kappa=(k_1, \dots, k_n)\in \mathbb{Z}^n$. Denote by $B_n$ the braid group on $n$ strings. Consider the braid on $n+1$ strings $\sigma_\kappa:=\sigma_1^{k_1}\...
- 10.5k
1
vote
0
answers
157
views
Khovanov $A_\infty$ algebra
Let $L$ be a link in $\mathbb{R}^3$, with $D, D'$ be diagrams in
$\mathbb{R}^2$ representing $L$. Khovanov constructed two graded
chain complexes $$C_{D} = (Ch_{D}, d_{D}) \quad C_{D'}=(Ch_{D'},
d_{D'}...
- 5,038
9
votes
1
answer
260
views
Physics application of Wilson surface observables
There is some work which generalises the usual Wilson loop in QFT to higher dimensions and constructs non-abelian Wilson surface functionals in the context of non-abelian gerbes.
It seems to me that ...
- 4,958
6
votes
1
answer
642
views
How to distinguish Pretzel links with the same coefficients?
Let $P:=P(a_1,\dots,a_n)$ be a Pretzel link ( https://en.wikipedia.org/wiki/Pretzel_link ).
For every permutation $\sigma\in S_n$ we can consider the link $$\sigma P:=P(a_{\sigma(1)},\dots,a_{\sigma(n)...
- 4,618
4
votes
1
answer
493
views
Cubic skein relations
please note that this question deals with undirected knots/links!
The most generic cubic skein relation for a knot polynome would be
$$S^2=wvS+w/S+w^2(u-v)I-u\cdot\infty$$
where $w^3$ is one ...
- 2,743
2
votes
1
answer
196
views
Knot concordance, hyperbolicity and amphichirality
Let $K_0$ and $K_1$ be two knots in $S^3$. We say $K_0$ and $K_1$ are concordant if there exists a smoothly embedded annulus $A \subset S^3 \times [0,1] $ such that $\partial A = -(K_0) \cup K_1$.
...
- 606
3
votes
0
answers
96
views
Kauffman bracket for Abelian anyons
The Kauffman bracket, defined here, assigns a polynomial in $A$ to any knot. (For concreteness consider the Kauffman bracket normalized so that the unknot is assigned $-(A^{-2}+A^2)$.) For certain ...
- 31
8
votes
1
answer
622
views
On trivial mapping class group of 3-manifolds
What are some examples of knots $K\subset S^3$ such that the mapping class group of $S^3_{1/n}(K)$ is trivial? I guess for hyperbolic knots with no symmetry in the complements are good candidate as ...
- 66.8k
2
votes
1
answer
195
views
Which hyperbolic fibered knots have monodromy with a single singularity?
The figure eight-knot has pseudo-Anosov monodromy with no singularity. I have read that the (-2,3,7)-pretzel knot has pseudo-Anosov monodromy with a single 18-prong singularity on the boundary of the ...
- 66.8k
7
votes
1
answer
419
views
Small knots becoming isotopic after connect sum
I am interested in the following situation: I have two codimension-2 knots $K_1$ and $K_2$ in $S^n$ and they are not isotopic. Furthermore, $K_1$ is not isotopic to the mirror image of $K_2$ and ...
- 19.2k
2
votes
1
answer
194
views
Determine if a closed braid is a link/unlink
I am relatively new to the world of braids/knots so really sorry if this question is simple. However, I am not able to find if there is any theorem/procedure that determines if a closed braid, given ...
- 73
3
votes
1
answer
166
views
Knot group of a field extension
Notation:
$L/K$, finite extension of global fields
$K^\times$, unit group of $K$
$L^\times$, units group of $L$
$\mathbb{A}_L^\times$, ideles of $L$
$N_{L/K}$, the norm map
The knot group of an ...
- 32.2k
7
votes
0
answers
248
views
Generating cycles inside Tits' graph of words for a positive braid
Let $Br_n$ be the braid group and consider words in its generators (not in the inverses). Two such words define the same "positive" braid if one can be obtained from the other by commuting ...
- 27.7k
5
votes
0
answers
118
views
Isotopy classes of $CP^1$ in 4-manifolds
Let $S_1$, $S_2$ be homologous embedded 2-spheres
in a compact smooth 4-manifold. Under which additional
conditions are they smoothly isotopic? I am interested
in the state of the art picture when $...
- 11.4k
3
votes
1
answer
156
views
Formula for the Casson invariant in terms of the linking form
The paper 'Trisections, intersection forms and the Torelli group' by Peter Lambert-Cole quotes the following formula for the Casson invariant of a knot $K$ in a homology $3$-sphere in terms of the ...
14
votes
4
answers
2k
views
Is there an algorithm for the genus of a knot?
A Seifert surface of a knot is a surface whose boundary is the knot. The genus of a knot is the minimal genus among all the Seifert surfaces of the knot. My question is, is any algorithm known to ...
- 66.8k
6
votes
1
answer
480
views
Fibered knots vs Heegaard genus
We call a knot $K$ in a 3-manifold $M$ fibered if $M\backslash K$ fibers over $S^1$ with fibers $\Sigma$ and such that $K$ is ambient isotopic to the boundary of the compactified fiber $\overline{\...
- 26.4k
1
vote
0
answers
37
views
Tetravalent graph invariant: Vassiliev in disguise?
Let's start with a virtual link, just that it has no over- and undercrossings, but simple nodes. Random example (A). For the virtual crossings, the usual laws hold (B). Also as usual, loose loops drop ...
- 4,699
2
votes
1
answer
270
views
Classification of congruent integer matrices
I am interested in the following question:
Let $A,B\in\text{Mat}(2n\times2n;\mathbb{Z})$ be two integer matrices with the property that $\text{det}(A-A^T)=1=\text{det}(B-B^T)$. Are there known ...
- 4,279
10
votes
4
answers
639
views
Is there any genus one knot other than $5_2$ with Alexander polynomial $2t^2-3t+2$?
It is known that genus one fibred knots are two trefoils and the figure-eight knot. Is there any characterization of the knot $5_2$? Specifically, is there any other genus one knot that shares the ...
- 101
8
votes
0
answers
162
views
Is the Lawrence–Krammer representation faithful, reduced modulo p?
It is well-known that the braid group $B_n$ is linear for every $n$ by the Lawrence–Krammer (or LKB) representation. It embeds $B_n$ faithfully into $\mathrm{GL}\left(\frac{n(n-1)}{2},\mathbb{Z}[q^{\...
- 113
19
votes
3
answers
1k
views
Does the super Temperley-Lieb algebra have a Z-form?
Background Let V denote the standard (2-dimensional) module for the Lie algebra sl2(C), or equivalently for the universal envelope U = U(sl2(C)). The Temperley-Lieb algebra TLd is the algebra of ...
- 4,618
3
votes
1
answer
309
views
A faulty proof that a Whitehead Double of a knot is smoothly slice
We denote the untwisted Whitehead double of a knot $K$ to be $Wh(K)$. As an example, here is the oriented Whitehead double of the figure eight knot:
Let us look in the neighborhood of the clasp:
...
- 10.5k
9
votes
4
answers
1k
views
Which knot complements are double branched covers?
Denote the double branched cover of a $2$-tangle $T\subset B^3$ by $\Sigma(T)$. Since $\partial \Sigma(T)$ is a torus, I wonder if anyone studied the question: which knot complements in $S^3$ are of ...
- 5,231
5
votes
0
answers
180
views
Lens space to lens space surgeries
Let $M_r(K)$ denote the slope $r$ surgery on a knot $K\subset M.$
Gordon conjectured and Kronheimer-Mrowka-Ozsvath-Szabo proved that if
$S^3_r(U)=S^3_r(K)$ for some $r$ then $K=U$ (the trivial knot).
...
- 5,231
3
votes
1
answer
128
views
Characterizing algebraic tangle by their double branched covers
Montesinos proved that the double branched cover $\Sigma(T)$ of an algebraic tangle $T$ in a $3$-ball is a graph manifold. I wonder if the converse true: Is $T$ algebraic if $\Sigma(T)$ is a graph ...
- 26.4k
2
votes
0
answers
107
views
General formula for a topologically slice odd pretzel knot
An odd 3-strand pretzel knot $K=P(p,q,r)$ has $\Delta_K(t)=1$ if $pq+pr+qr=1$. This fact, along with a theorem of Fintushel and Stern (every odd 3-pretzel knot with trivial Alexander polynomial is not ...
3
votes
1
answer
158
views
The same PD code seems to yield two different knot diagrams of the Hopf link
The PD code [(2, 3, 1, 4), (4, 1, 3, 2)] seems to map to a non-unique knot diagram. I can describe the following two Hopf links with different orientations with this same PD code. As I understand it, ...
- 1,617
5
votes
1
answer
270
views
Hyperbolicity of twist knots
In Example 1.4 of his book Invariants of Homology 3-Spheres, Saveliev noted that twist knots $K_n$ of type $(2n+2)_1$ are all hyperbolic. Here, $K_1$ is the figure-eight knot $4_1$ and $K_2$ is the ...
- 26.4k
7
votes
2
answers
2k
views
Ideals in the ring of single-variable Laurent polynomials with integer coefficients
I'm writing some software to automatically compute things like Alexander modules, Alexander ideals, Milnor signatures and Farber-Levine pairings for 1 and 2-knot complements. So among other things I'...
- 1,446
3
votes
0
answers
181
views
Reshuffling power series (aka Melvin–Morton expansion in knot theory)
I am struggling to understand a statement which follows from some change of variables in a power series. I think that the context does not really matter here, so I will put it at the end of the ...
- 571
7
votes
1
answer
293
views
Is it possible to separate two linked (geometric) circles in $\Bbb R^3$ by a set homeomorphic to the 2-sphere (with arbitrarily “bad” homeomorphism)?
$A$ and $B$ are two linked (geometric) circles in $\Bbb{R}^3$. (Let, for definiteness, both have radius = 1, the first lies in the $z=0$ plane and its center is the origin of coordinates $(0,0,0)$, ...
- 11.4k
1
vote
1
answer
2k
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?
- 4,618
14
votes
2
answers
737
views
Zeroes of the Alexander polynomial for achiral knots
Are there some known properties about the position (on the complex plane) of roots of the Alexander polynomial of achiral knots? They are shown as blue points in the following picture of roots for ...
- 66.8k
8
votes
0
answers
378
views
The figure eight knot complement in $S^3$
Recently I have been going through the book Hyperbolic Knot Theory by Jessica Purcell. Exercise 5.4 (on page 101) gives us a presentation of the fundamental group of $S^3 - K$ where $K$ is the figure-...
- 26.4k
48
votes
2
answers
2k
views
Can I wrap a suitcase with hair ties
Is there a nontrivial link in a big solid torus that is trivial in the ambient Euclidean space such that each circle is unknot and has a sufficiently small length?
It is motivated by a question that ...
- 356
8
votes
3
answers
2k
views
Is Murasugi's conjecture still open?
Normalize the Alexander polynomial (in $t$) so that the positive and negative exponents are balanced. For example in the Conway normalization, make the substitution $z = t^{1/2} - t^{-1/2}$. The ...
- 2,743
55
votes
3
answers
6k
views
Kirby calculus and local moves
Every orientable 3-manifold can be obtained from the 3-sphere by doing surgery along a framed link. Kirby's theorem says that the surgery along two framed links gives homeomorphic manifolds if and ...
- 2,743
6
votes
1
answer
445
views
4-manifold obtained from a ribbon disk exterior by attaching a 2-handle is simply-connected if its boundary is a homology sphere
I am reading Lemma 2.1 of this paper (https://arxiv.org/pdf/2012.12587.pdf) and I can't see why $W$ is simply-connected. Here is the situation:
Let $K$ be a ribbon knot in $S^3$; it bounds a ribbon ...
- 66.8k
1
vote
1
answer
234
views
Alexander polynomial of the pretzel knot $P(2m+1,2n,2k+1)$
Is there a (closed) formula for the Alexander polynomial of the pretzel knot $P(2m+1,2n,2k+1)$, $m,k\ge 0 , n \ge 1$ ?
- 4,618
8
votes
1
answer
315
views
A $k$-component link defines a map $T^k\rightarrow \operatorname{Conf}_k S^3$. Does the homotopy type capture Milnor's invariants?
A $k$-component link defines a map $T^k \rightarrow \operatorname{Conf}_k S^3$. Does the homotopy type of this map capture the Milnor invariants?
Some special cases:
$k=2$, no, it's null homologous, ...
- 11.9k
65
votes
4
answers
4k
views
Tying knots with reflecting lightrays
Let a lightray bounce around inside a cube whose faces
are (internal) mirrors.
If its slopes are rational, it will eventually form a cycle.
For example, starting with a point $p_0$ in the interior
of ...
- 149k
1
vote
0
answers
81
views
The position of complex points on the kiwi-graph of the Jones polynomial
Consider the "kiwi" graph below (the name came from the resemblance to the bird, the national symbol of New Zealand, the country of V. Jones) i.e., roots of the Jones polynomial for knots (...
- 71
12
votes
3
answers
1k
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 ...
- 26.4k
3
votes
0
answers
82
views
Integral homology $S^1\times S^2$'s smoothly bounding integral homology $S^1\times B^3$'s
Suppose we are given a compact orientable 3-manifold $M$ which is an integral homology $S^1\times S^2$. Then is there a way to determine whether $M$ bounds a smooth compact orientable 4-manifold which ...
- 475
13
votes
1
answer
1k
views
What is the state of research on finding all prime knots with 17 crossings?
In this 1998 journal paper, all the prime knots with 16 or fewer crossings are found (some of which were found earlier by others). There are over 1.7 million such knots. But the prime knots with 17 ...
- 26.4k