For questions on knot theory, the study of mathematical knots

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1answer
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Proving trefoil group is isomorphic to a fundamental group.

From this document exercise 2.13 states: Show that the trefoil knot group is isomorphic to the group $\langle a,b \space | \space a^3 = b^2 \rangle$. From Fact 2.9 (and also the fact that ...
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0answers
18 views

Linking circles inside an immersed surface

A smooth embedding $f : D \to \mathbb{R}^3$ can be isotoped to a canonical inclusion $D \hookrightarrow \mathbb{R}^3$. (This is part of a proof that only the unknot has the disk as a Seifert surface.) ...
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20 views

homeomorphic two spheres embedded in $\mathbb{R}^4$

Let $A$ and $A'$ be two annuli in $\mathbb{R}^3$. Suppose $A$ has $n$ half twists and $A'$ is with $m$ half twists, where $m$ and $n$ are even and $m\ne n$. It is clear that the surface resulting by ...
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0answers
22 views

Homeomorphic surfaces embedded in 4-space

A surface-knot is a closed connected surface embedded in the Euclidean 4-space $\mathbb{R}^4$. We consider the projection of the surface-knot into $\mathbb{R}^3$ with the singularity set contains of ...
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1answer
22 views

Non-trivial alternating with determinant 1

Suppose that $K$ is a non-trivial, alternating knot. Is it possible that $\det K = 1$ where $\det K=\Delta_K(-1)$? Using knotinfo, I checked that all non-trivial alternating knots with crossing less ...
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0answers
7 views

lower bound for integral of curve derivative

Let $\gamma_k:[0,1]\rightarrow\mathbb{R}^3$ be some arc-length parametrized $C^1$-curve, more specifically a knot. $\gamma: [0,1]\rightarrow\mathbb{R}^3$ is the limit of a sequence of such knots in ...
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1answer
21 views

Equivalence of two definitions of a knot

Let a knot in $\mathbb{R}^n$ be an embedding of $f: S^1 \to \mathbb{R}^n$ under the relation that two knots $f,g$ are equivalent if there is a 'non-crossing' homotopy of maps from $f$ to $g$ (i.e. ...
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1answer
20 views

Are 3#4 and 3*#4composite knots isotopic?

all I found (on wolfram) that there is one composite knot with seven crossings and that is the 3#4. But is this really equivalent to 3*#4 i.e. a composite knot of trefoil with opposite chirality and ...
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1answer
82 views

Homology of knot exterior in general manifold for not null-homologus knot

I was trying to figure out the homology of the knot complement when $K$ is not a rational null-homologous knot ($[K]\neq 0\in H_1(X_K,\mathbb Q)$). We then know by half-lives half dies that the ...
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1answer
70 views

Complement of a knot that *isn't* rationally null-homologous

Let $K$ be a knot in a closed, oriented 3-manifold $Y$. It is a standard fact that if $K$ is (at least rationally) null-homologous, then $H_1(Y-K;\mathbb{Z})$ is isomorphic to $H_1(Y;\mathbb{Z})\oplus ...
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0answers
14 views

Crossing change by Dehn surgery versus by projection

I read the following proposition in "Crossing changes" by Martin Scharlemann. A crossing change for a knot $K:S^1\to S^3$ with crossing disk $D\subset S^3$ can be obtained by performing Dehn surgery ...
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20 views

Surfaces with self intersection in 3-space

Let $F$ be a sphere in the Euclidean 3-space $\mathbb{R}^3$ With self intersection. Let $C$ be a double point circle in $F$. Then the double circle $C$ must bound a 2-disk in the standard sphere ...
3
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1answer
72 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}, ...
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1answer
26 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 ...
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2answers
87 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 ...
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83 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
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1answer
81 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
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0answers
27 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
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1answer
36 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
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2answers
112 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
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1answer
43 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 ...
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1answer
37 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 ...
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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 ...
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2answers
75 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
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0answers
70 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
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1answer
49 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, ...
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2answers
275 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
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1answer
55 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 ...
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0answers
23 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 ...
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3answers
224 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 ...
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0answers
38 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 $? ...
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0answers
123 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 ...
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0answers
59 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 ...
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0answers
26 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 ...
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1answer
227 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 ...
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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 ...
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0answers
27 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
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1answer
54 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
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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
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0answers
38 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 ...
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0answers
73 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. ...
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1answer
46 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_-) + ...
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1answer
50 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 ...
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0answers
67 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
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1answer
69 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 ...
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2answers
138 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
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1answer
54 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 ...
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0answers
92 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
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1answer
133 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 ( ...
2
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1answer
31 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 ...