For questions on knot theory, the study of mathematical knots

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3
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1answer
87 views

About the trefoil knot group and the connected sum of trefoil knots

Let $T_1$ be the standart trefoil knot, embedded in $\mathbb R^3$. Then, one can easily give a simple Wirtinger presentation of $\pi_1(\mathbb R^3 \setminus T_1)$ by $\langle a,b,c | a = bcb^{-1}, b=...
1
vote
1answer
31 views

proof that the Jones polynomial is an invariant

On page 153 of Colin Adam's knot theory book he describes the invariance of the X polynomial (a precursor to the Jones polynomial) under the first Reidemeister move (R1). In this process Adam's seems ...
2
votes
2answers
149 views

Irreducible link complements in $\mathbb S^3$

Let $L$ be an oriented link in the 3-sphere $\mathbb S^3$, consisting of two knot components, $\gamma_1$ and $\gamma_2$. I wonder now if the following is true: If the linking number $N(L)$ of $L$ is ...
6
votes
0answers
100 views

Algebraic topology needed for knot theory

Both Rolfsens Knots and Links and Lickorish knot theory require some knowledge of algebraic topology, what is a resource that covers the bare minimum I need to get through either of these? I am not ...
2
votes
1answer
87 views

Three dimensional definition of Alexander Polynomial

I heard there is a intrinsically three dimensional definition of the Alexander Polynomial for knots, where/what book offers an explanation of this? What kind of Math is needed? Elementary is better
0
votes
0answers
37 views

Hopf link and degree of a map

I'm considering a problem of computing the degree of a map $\varphi: S^{1} \times S^{1} \rightarrow S^{2}$ defined as $$\varphi(x, y) = \frac{\gamma_{1}(x)-\gamma_{2}(y)}{|\gamma_{1}-\gamma_{2}|}$$ $\...
2
votes
1answer
42 views

Knot diagram coloured with only one colour by any colouring

Let $K$ be a knot diagram of a knot in $\mathbb{R}^3$. Suppose $K$ admits only trivial colourings by any quandle (a colouring is said to be trivial if only one colour is used to colour the diagram). ...
2
votes
2answers
121 views

What is the abelianization of $\pi_1(\mathbb{R}^3\setminus k)$, where $k$ is a knot in $\mathbb{R}^3$

I don't understand what the three dimensional plane minus a knot really is. I would like to know this because I am studying for an exam and don't know how to work out these abelianization type of ...
2
votes
1answer
64 views

The composition of a tricolorable knot with another knot is always tricolorable

Prove that the composition of a tricolorable knot and another knot (except the unknot, whether tricolorable or not) is tricolorable. I understand that the composition of two tricolorable knots are ...
1
vote
1answer
55 views

The Jones polynomial of the connected sum of two links.

I've been working on some knot invaririants and specialy the Jones Polynomial. I was able to prove that $ V_{K_1 \# K_2} = V_{K_1} V_{K_2} $ for two knots $ K_1 $ and $ K_2 $ . So I found my self ...
2
votes
0answers
34 views

Show that $k$-colorings of a link are in bijection with homomorphism $\pi_1(\mathbb{R}^3\setminus L)\to D_k$

Here $D_k$ is the group of symmetries of a regular $k$-gon. $D_k$ has $2k$ elements, the $k$ rotations through multiples of $2\pi/k$ and the $k$ reflections. I think this is related to Wirtinger ...
2
votes
2answers
90 views

The first homology group $ H_1(E(K); Z) $ of a knot exterior is an infinite cyclic group which is generated by the class of the meridian.

I'm trying to solve the following exercice : Prove that the first homology group $H_1(E(K); Z)$ of a knot exterior is an infinite cyclic group which is generated by the class of the meridian. With ...
2
votes
0answers
71 views

The best definition of a knot?

I just started reading a book about knots and links and asked myself what is the best and most precise way to define a knot, just an embedding of $S^{1}$ into $S^{3}$ is not enough, right? Can ...
3
votes
1answer
91 views

Tight approximation of a Torus Knot length

Is there a simple formula for a tight approximation of the torus knot length ? (specifically a formula that does not involve integrals or any iterative procedures). The torus knot parameters are $(p, ...
11
votes
2answers
305 views

Laymans explanation of the relation between QFT and knot theory

Could someone give an laymans explanation of the relation between QFT and knot theory? What are the central ideas in Wittens work on the Jones polynomial?
2
votes
1answer
68 views

Perform 0-framed surgery, then remove neighbourhood of meridian. Is this the knot complement?

Let $K$ be a knot in $S^3$ and let $m$ be a meridian of $K$. Let $M_K$ be the 3-manifold obtained by performing 0-framed surgery on $K$. The meridian $m$ can also be viewed as a circle in $M_K$. Is $...
0
votes
0answers
26 views

Labeling the (p,q,r)-pretzel knot with transpositions from S4

For what values of p,q, and r, can the (p,q,r)-pretzel knot be labeled with transpositions from $S_4$? I'm kind of stuck on how to approach this one. All I've got so far is that there are six ...
3
votes
3answers
286 views

Knot Group and the Unknot

Hi I am stuck in trying to show that given a knot $K$ such that the knot group $\pi_1(K)=\mathbb Z$ then $K\simeq U$. I tried to use the fact that the infinite cyclic cover is the universal cover but ...
2
votes
0answers
45 views

Is it true that Morse function on non-trivial knot has at least 4 critical points?

I'm actually interested in the continuous case, for a non-trivial knot $S^1\rightarrow \mathbb{R}^3 $ is it true that the function $\sin(t)$ can not extend to a continuous function on $\mathbb{R}^3 $?
4
votes
0answers
128 views

How to classify this surface

I know that it should be either a sphere, torus, Klein bottle, real projective plane, or a connect sum of any combination of these, but I don't know the steps in identifying what kind of surface this ...
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vote
0answers
63 views

Surgery presentation for an abstract open book decomposition

Suppose $(\Sigma,\phi)$ is an abstract open book whose monodromy is expressed as a product of Dehn twists about boundary-parallel curves. Is there a standard way to produce a surgery presentation of ...
1
vote
0answers
28 views

Proving the Existence of n-linked knots

I was reading up on knots and links and came across: The Hopf Link: https://en.wikipedia.org/wiki/Hopf_link Solomon's Knot (Double Link): https://en.wikipedia.org/wiki/Solomon%27s_knot Which got me ...
1
vote
1answer
254 views

List of number of knots distinguished by Alexander polynomials

Is there a list of numbers of how many knots are disinguished by their Alexander polynomials? Up to certain crossing numbers, or for each crossing number individually. I`m trying to get a feel for how ...
25
votes
3answers
2k views

Why are all knots trivial in 4D?

A classical knot is defined to be an embedding $S^1 \to \mathbb{R}^3$ where $S^1$ is a 1-sphere or circle. Embeddings $S^1 \to \mathbb{R}^4$ are usually not considered knots because they are trivial ...
2
votes
0answers
32 views

Alexander-Conway polynomial of the sum of two knots

It is known that the Alexander polynomial of the sum of two knots $K=K_1\#K_2$ is equal to the product of the Alexander polynomial of the two summands $K_1$ and $K_2$. If the same true for the ...
3
votes
1answer
58 views

Planar Graph Isomorphism

In 1980, I. S. Filotti & Jack N. Mayer proved planar graph isomorphism testing could be done in polynomial time. Does anyone have an implementation of that? I have a few billion planar graphs ...
4
votes
1answer
43 views

Ways to link the unknot to a pole

Is there a way to show that the following ways of linking an unknot to an infinite horizontal pole are inequivalent? Perhaps the Wirtinger presentation would work, but I am not sure because of the ...
4
votes
0answers
51 views

Unique Conway notation for knots?

Is the Conway notation for a knot unique? Here are two rational tangles whose closures give the trefoil knot. However the Conway notation written for the trefoil knot is usually presented as 3 in ...
3
votes
0answers
75 views

How to draw the knot 2, -32, 41?

Hej, I have the following exercise: Draw the tangle 2, -32, 41 and the corresponding knots obtained by connecting the NW string to the NE string and the SW string to the SE string. (From C. C. ...
2
votes
1answer
74 views

Skein relation for the Jones polynomial - Example not working out

I've decided to learn some knot theory during this summer, using The Knot Book. Today, I showed that the Jones polynomial satisfies the Skein relation $$t^{-1}V(L_+) - tV(L_-) + (t^{-1/2}-t^{1/2})V(...
0
votes
1answer
60 views

Ambient Isotopy of Knots

Statement 1: Knots of opposite chirality have ambient isotopy, but not regular isotopy. Statement 2: We can then define two such knots to be equivalent if they are ambient isotopic, meaning that ...
1
vote
0answers
73 views

Where I can find the proof that- for every knot there is a Conway Notation?

At the end of The Knot Book - Collin Adams there is a list of knots. He has given a Conway Notation for each of those knots, from which I have assumed that every knot has a Conway Notation. Or for ...
2
votes
1answer
75 views

Which notation unambigously describes a knot?

For a chiral knot the Dowker notations for the knot and it's mirror image are the same. So the Dowker notation does not convey the information of chirality. I am wondering is there any notation that ...
1
vote
2answers
195 views

Which two knots are isotopic but not ambient isotopic?

Which two knots are isotopic but not ambient isotopic? How can we see that they are indeed not isotopic but not ambient isotopic?
3
votes
1answer
59 views

Mistake in the definitions of the linking number.

I am looking into the definition of the linking number. I've considered these two definitions. Consider a link $L$ with components $K_1$ and $K_2$, and respectively their embeddings $\gamma_1$ and $\...
5
votes
0answers
96 views

Twisted nail puzzle framed in terms of algebraic topology?

See here for a description of the puzzle: Twisted Nail Puzzle. My question is, can someone provide a description of the puzzle and its solution in context of the language of algebraic topology?
4
votes
1answer
171 views

Definition by degree and intersection number are equivalent (linking number). [repost]

I will here restate a question I asked earlier. It did not have much succes (probably by an incomplete introduction of the problem on my part). I am reading a paper by Ricca ( http://www.maths.ed.ac....
2
votes
1answer
32 views

What was the paper about flower-shaped knots?

I read a article about the possibility to bring knots in a "polar rose" projection, where there is only one crossing of higher multiplicity. The overcrossing/ undercrossing information is thus more ...
8
votes
1answer
461 views

Equivalence of knots: ambient isotopy vs. homeomorphism

I am looking into knot theory and have found two different definitions stating that two knots $K_1$ and $K_2$ are equivalent, namely the concept of an ambient isotopy: These two knots are ambient ...
3
votes
1answer
68 views

Alexander polynomial of unknot without Fox calculus or infinite cyclic cover

As explained in Lickorish`s book "Introduction to knot theory", one can define the Conway-normalized version of the Alexander polynomial by the determinant of certain sum of Seifert matrix plus ...
2
votes
1answer
52 views

Definition of a rim torus

We know a torus is $S^1 \times S^1 =T^2$. We know a solid torus is $D^2 \times S^1$ whose boundary is a torus $S^1 \times S^1 =T^2$. What is the definition of a rim torus?
4
votes
1answer
79 views

When does $\pi_1(\Sigma)$ inject into $\pi_1(S^3 \setminus \Sigma)$?

Here's a fun fact from knot theory: $\quad$ If $\, \Sigma$ is a minimal-genus Seifert surface for a knot $K$, then $i_*:\pi_1(S^3 \setminus \Sigma) \to \pi_1(S^3 \setminus K)$ is injective, where $i:...
4
votes
1answer
93 views

Is there a one to one correspondence between Jones' polynomials and knots?

I know Jones' polynomial is a knot invariant. By using knot invariant like p-coloration one can only say whether two knots are different but not whether they are the same. So it is like injective ...
2
votes
1answer
71 views

Knot invariants that discern prime and composite knots.

Is there a list of knot invariants that can tell whether or not a knot is prime? Or at least partially so? i.e. invariants that have one or more of the following properties: (a) The invariant has a ...
3
votes
1answer
100 views

Remove one ring of Borromean rings in 3-sphere: linked or unlinked?

We know Borromean rings in a 3-sphere $S^3$ can be unlinked if we remove one of the three rings. Here let us consider a slight different procedure. If we remove the neighbored solid torus $B^2 \times ...
2
votes
2answers
175 views

Why is the Hopf link the only link with knot group $\mathbb{Z} \oplus \mathbb{Z}$?

We can use the Loop Theorem to show that if $\Sigma$ is a minimal-genus Seifert surface for a link $L$, then $\pi_1(\Sigma)$ injects into the knot group $\pi_1(S^3 \setminus L)$. An orientable ...
2
votes
0answers
82 views

Equivalence (or not) of two Artin/Fox wild arcs

The repeating patterns in the wikipedia articles on wild arcs and wild knots seem to me to be not continuously deformable to each other. Is this true? For clarity, here is my diagram of the repeating ...
1
vote
0answers
95 views

Solving integrals with delta function constraints

What is the best way to solve integrals which use delta function constraints/restrictions? For example if I have the integral $\int_V \exp({\frac{-1}{a}\Sigma_{i=0}^N(t_{i+1}-t_i)^2})\delta({t_{N+1}-...
2
votes
1answer
121 views

Higher homology groups of infinite cyclic cover

Prove that all homology groups of the infinite cyclic cover of a knot complement are trivial except $H_1$. I've posted an answer below using Mayer-Vietoris. If you know of other arguments, please ...
3
votes
1answer
51 views

Showing that gluing two knot exteriors together contains subgroups isomorphic with the knot groups.

I'm working through Rolfsen's "Knots and Links" and section 9D exercise 10 has me stumped: Let $K_1$ and $K_2$ be knots in two separate copies of $S^3$ with respective meridians $m_1$ and $m_2$ and ...