# Tagged Questions

For questions on knot theory, the study of mathematical knots

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### Geometric way to view the truncated braid groups?

This is perhaps a vague question, but hopefully there exists literature on the subject. The question is motivated by an answer I gave to this question. I also asked a related question on MO, although ...
244 views

### How did Chern pictured the first Chern number?

The first Chern number $\cal C$ is known to be related to various physical objects. Gauge fields are known as connections of some principle bundles. In particular, principle $U(1)$ bundle is said to ...
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### Why is the Mazur swindle named so?

Often results or techniques in mathematics are called 'theorems'. Sometimes they are called 'tricks'. In no other context have I seen a result called a 'swindle'. Is there a historical reason for this ...
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### Is “slightly deform” a well defined concept in mathematical proof?

In topological proofs the phrase "slightly deform" is widely used. To me, although I can accept the idea intuitively, the phrase "slightly deform" does not sound like a strict mathematical concept. ...
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### Has knot theory led to the development of better knots?

Knot theory was likely originally motivated by the study of real-world knots such as these: Indeed, mathematical knot tables to this day look not too dissimilar from the familiar "age of sail"-...
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### Definition of Reshetikhin-Turaev TQFT

I am studying Reshetikhin-Turaev TQFT. In their paper or in the book " Quantum invariants of knots and 3-manifolds", they first define an invariant $\tau(M)$ for a closed orientable 3-manifold $M$ and ...
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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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### 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?
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### Can information about a knot be recovered from the Jones Polynomial?

Suppose we know the Jones polynomial of some knot, but maybe not specifically which knot. Can any information about the knot be recovered just by knowing its Jones polynomial? Say, for example, the ...
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### if a circle is deformation retract of a manifold; then is there a knot deformation retract to an embedded M in 3-space

Let M be a metric space such that the circle is deformation retract of M. If a circle is embedded in 3-space, we obtain a knot. Suppose the space M is embedded in 3-space and its projection image in ...
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### On the definition of Fox derivative

I am reading An Introduction to Knot Theory by W.B. Raymond Lickorish. In Chapter 11 the motivation for the Fox derivative is mentioned. I understand why the contribution of the occurrence of $x_j$ in ...
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### Prove that $S^{1}$ unknots in $\mathbb{R}^{4}$

The definition is, X unknots in Y if any two embeddings are equivalent. How do you show $S^{1}$ unknots in $\mathbb{R}^{4}$ and in general, $S^{n}$ unknots or knots in $\mathbb{R}^{m}$? The solution ...
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### Alexanderpolynomial of connected sum via Fox calculus and Wirtinger presentation

Hello :) i have just reading the question "How to compute the Alexander polynomial of general torus knot" and i was suprised how strong it works if someone have a difficult question. I am also very ...
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### some help on the group of unknotted

Show that the group of the unknotted $K=\{(z_z,z_2)\in \mathbb{S^3} : |z_1|=1 \}$ is infinite cyclic. where $\mathbb{S^3}$ is to be considering as the unit vectors in $\mathbb{C^2}\cong \mathbb{R^4}$. ...
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### does a method exist to distinguish two component link consisting of just two unknots from an unlink?

Clearly, linking number is not enough as there are links like whitehead. There is the enhanced linking number based on conway polynomial that can distinguish whitehead (and infinite family of such ...
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### $3\tau(K_1$#$K_2)$=$\tau(K_1)\tau(K_2)$

Suppose we have two knots $K_1$ and $K_2$. Then look to the connected sum of $K_1$ and $K_2$ denoted by $K_1$#$K_2$ (defined for knots). Suppose $\tau$ is the number of $3$-colourings (definition for ...
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### 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 ...
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### Linking integral unchanged over continuous deformations

Say we first have two curves, $C_1$ and $C_2$ which are knotted together. Let $C_2'$ be a continuous deformation of $C_2$ such that $C_2$ does not cross $C_1$ as it is deformed into $C_2'$. How ...
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### Can the HOMFLY polynomial be obtained from the Kauffman Polynomial for torus knots?

This is essentially a yes/no/reference request question. Let me first just ask my question: Is there a known relationship between the HOMFLY and Kauffman polynomials of torus knots? In particular, ...
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### L(q,p)# L(p,q) = surgery on pq-torus knot complement

I am trying to solve one of Rolfsen exercice. That is prove that the connected sum $L(q,p)\# L(p,q)$ can be obtain by surgery on the complement of the pq-torus knot in $S^3$. I am doing it using ...
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### If a strip of paper is knotted into an open trefoil, what is the linking number?

It is assumed that the paper strip is knotted into an open trefoil (forming a pentangle) that lies flat on the table, and that the two ends of the paper strip are continued up to spatial infinity. (...
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### Homology, addition of homology classes in construction of Poicare Sphere

I am working through Greenberg and Harper, Lecture notes on Algebraic Topology, and I am having trouble with one exercise. I have spoken with a professor and he encouraged me to ask here or look for ...
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### How is the Alexander polynomial computed from the Alexander quandle?

I have computed the Alexander Polynomial through the skein relation but sources such as Wikipedia and nLab say: The Alexander quandles are also important, since they can be used to compute the ...
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### The Crossing Number of a family of graphs which contain the complete bipartite graphs.

Let $p,q$ and $r$ be positive integers greater than $0$ with $q\neq r$. Suppose that $H$ is a finite connected graph without loops or multiedges on $p$ vertices with $q$ vertices of degree $r$, $r$ ...
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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### colouring knot diagram and its mirror by the same colouring

Let $K$ be a knot diagram coloured by any quandle $X$. Let the colouring used be $C$. Reverse the orientation of $K$ to obtain the reverse of $K$, denoted by $-K$. Then is it possible to colour $-K$ ...