Questions tagged [knot-theory]

For questions on knot theory, the study of mathematical knots and their properties.

Filter by
Sorted by
Tagged with
2
votes
0answers
25 views

Higher homotopy groups of a string complement in a cylinder.

We call the image of a smooth embedding $f:\coprod_{i=1}^n [0,1]\times\{i\} \hookrightarrow \mathbb{D}^2 \times [0,1] $ a string if $f(0,i)$ and $f(1,i)$ have the second coordinates $0$ and $1$ ...
1
vote
0answers
11 views

Simple examples of colored HOMFLY polynomial?

I'm trying to find some simple examples of using the colored HOMFLY polynomial and was wondering if anyone could give me a reference. I really only need an extremly basic definition of the colored ...
1
vote
0answers
16 views

Where can I get an introduction to the Colored HOMFLY Polynomial?

Where can I get an free introduction to the Colored HOMFLY Polynomial? It would be great If I could get an exposition of the invariant with regards to rational links, but I'm not holding my breath. I ...
2
votes
1answer
62 views

Is the Mazur swindle a supertask?

The Mazur swindle can be used to show that any knot with an inverse under connected sum must be isotopic to the unknot. The isotopy, as I understand it, involves instantiating a pair of the knot and ...
1
vote
1answer
26 views

Unusual skein relation in HOMFLY polynomial

If I take the HOMFLY(PT) polynomial defined by $$l \,P(L_+) + l^{-1}\,P(L_-) + m\,P(L_0) = 0,$$ I have looked at expressions of the form (knots that are the same except inside a small disk, where ...
1
vote
2answers
50 views

Given a sequence of head to tail vectors forming a closed loop, how can I determine if they form a knot? [closed]

Consider if we have some sequence of vectors placed head to tail which form a closed loop. How can one determine whether they form a loop? We assume that it is given that the vectors close, that is ...
2
votes
0answers
125 views

Distortion of the Unknot

In Mikhail Gromov's "Filling Riemannian Manifolds" he defines the distortion of a knot $K$ embedded in $S^3$ as $$\delta (K) := \inf_{\gamma \in K} \sup_{x,y \in \gamma} \frac{d_{\gamma}(x,y)}{||x-y||}...
0
votes
0answers
20 views

Parametric curve end tip orientation

If I have a parametric curve equation, say for a trefoil ($3_1$) knot $x = 15(t^3-3t)$ $y = 15(-t^4+4t^2)$ $z= -15(0.2t^5-2t)$ Suppose I plot this only between t=(-2,0). Say the coordinate at the tip ...
1
vote
1answer
24 views

Tying a portion of a knot diagram

I am having some trouble with a paper by Livingston (Infinite Order Amphicheiral Knots, Algebraic and Geometric Topology 1, 2001, 231-241). He starts with a knot $K$, and constructs a new knot "...
0
votes
1answer
38 views

A knot $K = K_1 \# K_2$ is alternating if and only if $K_1$ and $K_2$ are alternating.

Is a knot $K = K_1 \# K_2$ is alternating if and only if $K_1$ and $K_2$ are alternating? In particular, I'm interested in the following direction: If $K_1 + K_2$ is an alternating knot, are both $...
0
votes
0answers
14 views

Does the link with trivial knot group trivial? [duplicate]

I know that if a the knot group of a classical knot is isomorphic to the infinite cyclic group, then the knot is unknotted. How about the link, is this result also valid for links. In other words, if ...
1
vote
2answers
67 views

How to see that the figure 8 knot has genus 1?

I am reading the knot book from Colin Adams, and after he discussed how one may construct a (Seifert) surface whose boundary is the knot, he went on and stated that the figure 8 knot has genus 1 (that ...
0
votes
0answers
5 views

Does a succession of knot slicings create a 4 dimensional knot?

Consider the unknot floating about and then it curls up and a line passes through another line and it becomes a trefoil knot. Then a line passes through a line again at it becomes the unknot. If we ...
0
votes
0answers
32 views

Knot polynomial of composite knot

This is a question from “An introduction to knot theory” $(GTM 57)$ If we tie two knots on the same piece of string, the result is called a composite knot. Prove that the Alexander ...
4
votes
0answers
64 views

Do there exist non-trivial knots whose Jones polynomial is a unit?

Question: It's an open problem whether or not the Jones polynomial distinguishes the unknot from all other knots. That is, the following problem is unsolved. Does there exist a knot $K$ which is ...
0
votes
0answers
35 views

Wild knots and connected sum

I am a little bit confused. If I take the next connected sum K 1#K 2#K 1#K 2.... Then why it's well defined? I red a proof that used the fact that if K 2#K 1 is the unknot (hence also K 1#K 2) so: K ...
1
vote
1answer
37 views

Reference request for surgery on knots

I've seen this article https://www.math.cuhk.edu.hk/~ztwu/JonesCosmetic.pdf on the Jones Polynomial and Cosmetic Surgery and I've looked at the Wikipedia entry on Dehn surgery as well. My background ...
1
vote
1answer
35 views

Is the linking number via Seifert surfaces well defined?

Let $i(K, F_L)$ be the signed count of intersections of an oriented knot $K$ with a Seifert surface $F_L$. (That is, $F_L$ is an oriented compact surface with boundary $L$ for some knot $L$.) I want ...
0
votes
1answer
33 views

Express the Klein bottle as the cofibre of a map $\kappa$ between (wedges of ) copies of $S^1.$

Express the Klein bottle as the cofibre of a map $\kappa$ between (wedges of ) copies of $S^1.$ Describe the map explicitly. Could anyone help me in finding this map please?
0
votes
0answers
24 views

Topological equivalence of 2-tangles

I am trying to better understand topological equivalence of tangles, in particular what deformations are allowed and which are not. Here is the definition of tangle that I'm using from wikipedia. ...
0
votes
0answers
52 views

Knots and Embedding

What is the different between a smooth embedding and a homeomorphism? I read a little about knots and I do not understand how a knot can be an embedding of the unit circle but not homeomorphic to it?
1
vote
1answer
63 views

Show that a retract of a cofibration is also a cofibration.

Here is the question: Suppose that $g: A \rightarrow C $ is a retract of $f: B \rightarrow D.$ Show that if $f$ is a cofibration, then so is $g.$ Could anyone help me in answering this question, ...
1
vote
0answers
76 views

Problem 22.39(b) in “ Modern classical homotopy Theory ” by Jeffery Strom on pg.511.(u is the natural transformation of cohomology theories.)

Here is the problem: Suppose $R$ is a field. (a) Show that $h^{n}(?) = Hom_{R}(H_{n}(?; R), R)$ is a cohomology theory defined on (at least) the category of finite CW complexes. (b) Show that $u$ ...
1
vote
1answer
21 views

Upper bound on the genus produced by Seifert's algorithm

Given a knot $K$ we can apply Seifert's algorithm to produce a surface whose boundary is $K$. The genus of this surface is not necessarily minimal. Is there an upper bound on the genus of the surface ...
0
votes
1answer
34 views

Terminology for two trefoil knots linked together

Is there a term describe the link formed by two trefoil knots (or any prime knots, I suppose) linked together like so? Also, does that particular link correspond to a prime link?
2
votes
1answer
68 views

Unknot: embedded vs immersed bounding disk

Suppose that we have a knot $K\subset \mathbb{S}^3$. If $K$ bounds* an embedded 2-disk then $K$ is the unknot. But what happens if $K$ bounds an immersed 2-disk? The immersed disk generically will ...
2
votes
0answers
25 views

Slice disks with only a minimum are bounded by the unknot

Let $D\subset B^4$ be a properly embedded 2-disk in the 4-ball. Suppose the the radial function on $B^4$ has only a critical point on $D$. Then is $\partial D\subset \partial B^4=S^3$ the unknot? In ...
0
votes
0answers
21 views

Is the connected sum of links well-defined?

Suppose you have 2 links L1 and L2, is the connected sum of these 2 links L1 # L2 well-defined? why or why not? Thanks in advance for the help
1
vote
1answer
62 views

Describe a map $f: S^2 \times S^2 \rightarrow S^4$ such that $f^{*}$ is an isomorphism.

Here is the question: Describe a map $f: S^2 \times S^2 \rightarrow S^4$ such that $f^{*}: \tilde{H^4}(S^4, \mathbb{Z}) \rightarrow \tilde{H^4}(S^2 \times S^2, \mathbb{Z})$ is an isomorphism. Does ...
1
vote
1answer
33 views

Does every pairing of odd and even number in $[n]$ correspond to some alternating knot?

I am reading about the Dowker notation in "The Knot Book" By Collins Adam. There's a link to the book below http://people.math.harvard.edu/~ctm/home/text/books/adams/knot_book/knot_book.pdf An ...
0
votes
1answer
80 views

Is there a classification of all classifications in mathematics? [closed]

Not sure if this question is too general for math.stackexchange but I will try. I find the classification of finite simple groups fascinating. That we have sorted all possible finite symmetries into ...
0
votes
1answer
23 views

Existence of Seifert surfaces in Ranicki

For knots in $S^3$, I am used to seeing Seifert's algorithm used to prove the existence of Seifert surfaces. However, In Ranicki's book on knots, High Dimensional Knot Theory, he gives a different, ...
6
votes
0answers
94 views

Mayer-Vietoris sequence to find first homology for the double cover of complement of trefoil knot

I apologize in advance about the pictures...I am not sure how to properly crop/rotate them. Suppose $K: S^1\to S^3$ is a map whose image is the trefoil knot. Let $X=S^3\setminus K$ (identifying $K$ ...
7
votes
1answer
115 views

Does the fundamental group detect the unlink?

It is known that the unknot is the only knot whose complement has fundamental group $\mathbb{Z}$. Does this fact generalize to links? That is, suppose that $\ell= \ell_1 \cup \dots \cup \ell_n$ is ...
-3
votes
1answer
53 views

Why Eilenberg Maclane spaces $K(G,n)$ are $(n-1)$ connected?

Why Eilenberg Maclane spaces $K(G,n)$ are $(n-1)$ connected? could anyone explain this for me please?
2
votes
0answers
28 views

Surfaces in $S^4$ with one bounding a Seifert solid disjoint from the other

Let $F_1, F_2 \subset S^4$ be two smoothly embedded disjoint closed orientable surfaces. Assume that $H_1(F_1; \mathbb{Z}) \to H_1(S^4 - F_2; \mathbb{Z})$ is the 0 map - does it follow that $F_2$ ...
3
votes
1answer
56 views

How to show that $\pi_1(\mathbb{R}^3\setminus k_{2,3}\#-k_{2,3})\not\cong\pi_1(\mathbb{R}^3\setminus k_{2,3}\# k_{2,3})$?

How to show that $\pi_1(\mathbb{R}^3\setminus k_{2,3}\#-k_{2,3})\not\cong\pi_1(\mathbb{R}^3\setminus k_{2,3}\# k_{2,3})$ by their presentations? Here $k_{2,3}$ stands for the trefoil knot. $k_{2,3}\#...
0
votes
1answer
31 views

computing the Kauffman bracket with the given relation

My Problem: Use the relation to compute the bracket of the link diagram $D_n$ with $n$ components: My attempt: It seems to me that raising the given equation to the $n^{th}$ power is the most ...
1
vote
0answers
39 views

Algebraically slice implies slice

I am studying an article by Livingston ("New examples of non-slice, algebraically slice knots", Proceeding of the AMS, 2001) about an example of an infinite class of knots which are algebraically ...
1
vote
1answer
51 views

Compute $H_{*}(M, \mathbb{Z})$ of gluing 2 solid tori.

Let $M$ be the space obtained by gluing two solid tori $D^2 \times S^1$ and $S^1 \times D^2$ together via the identity map of their boundaries. Compute $H_{*}(M, \mathbb{Z}.)$ A Hint: Use Mayer ...
1
vote
1answer
38 views

Conway Notation Question For a Knot Diagram

my question is fairly simple. I have been staring at this knot diagram for a while and cannot understand why its Conway Notation is 2 -2 2 -2 2 4 (according to Adam's The Knot Book). The way I ...
0
votes
1answer
47 views

Alexander polynomial of any knot evaluated at 1 is $\pm$ 1

I'm supposed to prove for a knot theory homework assignment that the Alexander polynomial of any knot (as opposed to link) is $\pm1$. From examples, I'm pretty convinced that this is true, but I have ...
1
vote
1answer
18 views

A regular Morse map $f:C_K\to S^1$

A circle-valued Morse map $f:C_K \to S^1$ on the complement of a knot $K$ is said to be regular if there is a $C^\infty$ trivialisation $\Phi : N(K)\to K\times B^2(0,\epsilon)$ of a tubular ...
1
vote
1answer
42 views

Why do we study representation of knot groups?

In my independent study class in Knot Theory, my professor said that solving the relation in knot groups is an undecidable problem, and he said we study representations of knot groups because then ...
3
votes
1answer
22 views

Homotopy rel boundary of disks in knot complement $\times I$

Let $K$ and $J$ be disjoint knots in $S^3$ and let $D_1$ and $D_2$ be two immersed disks in $(S^3 - K) \times I$ with boundary $J$. Is it true that $D_1$ and $D_2$ are homotopic through a homotopy ...
3
votes
1answer
88 views

A wild knot and its complement

I have been reading about wild knots in $\mathbb{R^3}$ that have nonsimply connected complements. I'm a bit confused here and I added a picture of a wild curve with two circles in its complement. Is ...
0
votes
0answers
58 views

Baffling bit of notation

So I'm reading a paper about deriving polynomial invariants for links in an arbitrary surface, and I'm stuck on some under explained notation. The full quote states that something is a 'right module ...
3
votes
1answer
119 views

Conjugacy classes with linking number 1 in the complement of a knot?

Let $K$ be a knot in $S^3$ and let $lk: \pi_1(S^3 - K) \to H_1(S^3-K;\mathbb{Z})$ denote the abelianization map. Let $1 \in H_1(S^3-K;\mathbb{Z})$ denote a choice of generator. Is it possible that ...
1
vote
0answers
18 views

Equivalence of two elements in $\pi_2(BX)$, where $BX$ is the rack space corresponding to the quandle $X$.

I am reading the proof of Lemma 4.1 given in the chapter "Some of Quandle Cocycle Invariants of links" of the book "Quandles and Topological Pairs" by "Nosaka." Before coming to the question, I have ...
0
votes
0answers
25 views

Proving that Hopf link is fibered

What’s the simplest way to show that the Hopf link is a fibered knot with the Hopf band as fiber surface and that its monodromy is a Dehn twist along the core of the Hopf band?

1
2 3 4 5
18