For questions on knot theory, the study of mathematical knots

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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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1answer
16 views

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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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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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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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
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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1answer
83 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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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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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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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 ...
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55 views

double decker set in a surface-knot

Surface-knot is an embedded surface in $\Bbb{R}^4$. Project the surface in $\Bbb{R}^3$ gives the surface diagram with set of singularity points consists of double points, triple points and branch ...
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1answer
73 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
88 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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1answer
66 views

Are there any combinatorial studies of Kirby calculus?

All of the other diagrammatic calculi I know of can be utilised with basically just combinatorial knowledge - for instance calculating knot and link polynomials. Are there similar combinatorial ...
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1answer
165 views

Are there Kirby diagrams for manifolds with boundaries?

There are Kirby diagrams for 3- and 4-manifolds which consist of framed links corresponding to 1- and 2-handles attached to a single 0-handle. Any such diagram will give a unique closed manifold since ...
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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 ...
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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 ...
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1answer
141 views

Showing every knot has a regular projection using differential topology

Can we use differential topology to prove that every smooth knot has a regular projection? Here is some background: Let $\gamma : S^1 \rightarrow \mathbb{R}^3$ be a smooth unit-speed imbedding. For ...
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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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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). ...
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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 $$\varphi(x, y) = \frac{\gamma_{1}(x)-\gamma_{2}(y)}{|\gamma_{1}-\gamma_{2}|}$$ ...
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1answer
38 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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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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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 ...
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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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1answer
60 views

Knots which are composed of several strands

In a math textbook and this article in NRICH, some problems deal with a special kind of knots: those which are formed from several strands: The problems ask if a given knot can be formed from just ...
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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 ...
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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?
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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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0answers
105 views

Find the fundamental group and the Alexander polynomial

I would like to find the Alexander polynomial of the link $L$, described below. Let $K(q,r)$ be the $(q,r)$-torus knot embedded on a torus $V$. Inside the torus $V$, consider a smaller solid torus ...
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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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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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1answer
240 views

Will a knot tied in a hanging, frictionless rope slip out under the force of gravity?

I am overall just curious about what keeps knots where they are in a rope. Another related question you might be able to answer is: What happens if you tie a bowline on the bight in a frictionless ...
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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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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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3answers
54 views

6-coloring of a knot

According to this page this knot should be 6-colorable (question 6): But I couldn't find an explicit coloring, which makes me think that the claim in the parantheses is not true. Can you find one? ...
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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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1answer
103 views

Recovering knot crossing orientations from a Gauss code or Dowker notation

Some common representations of knots do not directly give the sign/orientation of each crossing. For instance, the trefoil knot has Gauss code -1, 3, -2, 1, -3, 2 and Dowker-Thistlethwaite code 4 ...
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3answers
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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 ...
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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 ...
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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 ...
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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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80 views

Classify knots in a closed bead-spring like polymer simulation

I'm trying to detect the crossing number (or another knot invariant) of a simulated polymer. A polymer is a closed bead spring, which means that it is represented by a set of points connected by ...