For questions on knot theory, the study of mathematical knots

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Alexander-Conway polynomial of an unlinked knot…

I had asked this elsewhere earlier in the week but I decided I am more likely to get an answer here: Is it true that for all unlinks, the Alexander-Conway polynomial is equivalent to 0? It seems ...
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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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On the definition of positive and negative crossings

In the definition of the Kauffman bracket, we have to resolve a crossing in two ways. The way to resolve these crossings involve distinguishing whether the crossings are positive or negative. But ...
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Kauffman Bracket

Can anyone explain or direct me to: . What are the x and y's? Is x cutting the loop? And for the last picture, I don't understand how the Reidemeister moves have been applied there. I would much ...
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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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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 ...
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William Thurston's 'Knots to Narnia'

http://www.youtube.com/watch?v=IKSrBt2kFD4 Above is the youtube link to the short video called 'Knots to Narnia'. (9 mins) While learning knot theory, I found this interesting video. In the video, ...
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Burau representation and the Jones polynomial

It is well known that if $K$ is the closure of a braid $f$ in the braid group $B_n$ then, up to some overall factor, the Alexander polynomial $\Delta_K(t)$ of $K$ is given by \begin{equation} ...
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Knots as boundaries

Boundary of a 2-manifold is a closed curve (or a set of closed curves), so I was thinking of reversing this process. In 2D space, a simple closed curve in a plane can be thought of as a boundary of ...
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minimal diagram of a knot and the crossing changes

Given any knot diagam, we can deform it into unknot by switching over arc and under arc of some crossings in the diagram. My question is the following: Let K be a knot a diagram of a knot, suppose ...
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Obtaining Wirtinger presentation using van Kampen theorem

Hatcher's Algebraic Topology, section 1.2, problem #22 describes an algorithm for computing the Wirtinger presentation of the complement of a smooth or piecewise linear knot K in $\mathbb{R}^3$: ...
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Representations of knot groups

Recently, I was studying the knot group and I want to learn some more material about it (e.g. its representations). "Knots" by Burde and Zieschang discusses some material but it is not entirely ...
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Trapping a sphere in a trefoil knot

Is it possible to trap a sphere in the center of a trefoil knot? It seems like with three points of contact it should be possible. Admittedly not big into math, but a craftsman who loves trapping ...
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smoothing a triple point by 1-handle addition

Given a surface diagram of a surface knot, we can attach 1-handle between the bottom and the middle sheet or the top and the middle sheets, in both cases a triple point is smoothed. My question is ...
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Category theory for knot theory

Where would I start to learn category theory for its use in knot theory? I have a background in physics and Ive read Adams Knot book. I know nothing about category theory. Eventually I want to learn ...
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2answers
73 views

Is there some knot theory behind the Mobius donut?

I was watching this video by Numberphile where a professor cuts a bagel into two interlocking pieces. Is this a torus knot or torus link? I'm trying to interpret in terms of $(p,q)$-torus knots Torus ...
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How do we define the mirror image of a knot in general 3-manifolds

How do we define the mirror image of a knot in general oriented 3-manifolds ? For instance for a knot in an irreducible integer homology sphere.
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Legendrian isotopy

I think the definition of Legendrian isotopy is that you can find an isotopy of the ambient manifold such that it takes a Legendrian knot to another through Legendrian knots. What I cannot figure out ...
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Probability that two circles in space are linked

Let $C_0$ be a circle centered on the origin, and $C_1$ a circle centered on $(1,0,0)$, center distance of $1$. Q1. If both $C_0$ and $C_1$ are randomly oriented and have the same radius $r ...
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Bridge index of Pretzel link

I was studying the bridge number of various kinds of links.I have heard there is some correlation between the bridge index of a Pretzel link and it's representation.Can anyone please explain it or ...
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2answers
62 views

is knot type invariant under diffeomorphism?

Is it possible to have a diffeomorphism of $R^3$ which changes the knot type, for instance the image of a trivial knot is a trefoil knot?
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Rolfsen exercise, chord theorem

Here's a problem from Rolfsen's Knots and Links that has me scratching my head: Show that there is always a counterexample to the "chord theorem" if $n$ is not an integer. [Hint: In attempting to ...
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1answer
43 views

How do we know how many prime knots there are for a specific number of crossings?

For instance, on wikipedia it says there are 7 prime knots with 7 crossings. How do we know there isn't an 8th prime knot knot with 7 crossings that we haven't yet discovered?
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A comment on a proof of equivalent knots

The following theorem and its proof is from A First Course in Algebraic Topology By Czes Kosniowski pp. 219-220 ...
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Closed regular neighborhood

I would like to understand the following sentences. Let $L$ be a framed link in the three dimensional sphere $S^3$. Suppose $L$ has $m$ components $L_1, \cdots, L_m$. Let $U$ be a closed regular ...
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80 views

How many ways are there of tying a tie?

I am sorry if this is useless. I have read in a newspaper that mathematicians have found the number of ways a tie can be tied. How could such a problem be solved? I'm asking out of curiosity.
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Why a torus knot is a prime knot?

Why a torus knot is a prime knot?
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Finding the jones polynomial of the following knot

I am struggling with finding the Jones polynomial of the following knot using only the sklein relations I am also asked if it is Isotopic to its mirror image. I tried using the above relation, ...
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161 views

two interlocked circles are homeomorphic to two noninterlocked circles

This is what I learned from here the post: two interlocked circles are homeomorphic to two noninterlocked circles, thus they (two interlocked circles and two noninterlocked circles) are homotopic ...
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How to distinguish between knots and links based on knot diagrams/projections

I'm interested in the distinction between knots and links in $\mathbb{R}^3$/$S^3$. In particular, is there an algorithmic way (as in not by sight/intuition) that we can examine the arcs and crossings ...
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Homology of knot complement

I was told in a topology class that if $Y$ is a closed $3$-manifold and $K$ is a null-homologous knot in $Y$, then $H_1(Y- \nu(K)) \cong H_1(Y) \oplus \mathbb{Z}$. I'm trying to prove this ...
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third quandle homology group $H_3^Q(R_3;\mathbb{Z}) \cong \mathbb{Z}_3$

It is known that the third quandle homology group $H_3^Q(R_3;\mathbb{Z}) \cong \mathbb{Z}_3$. Each colouring $C$ of a knot diagram by a dihederal quandle $R_3$ induces a homomorphism $f:C \rightarrow ...
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fibered knots in $ S^3$

Given a fibered knot $k$ in $S^3$, we have the decomposition of $S^3$ as union of $M$ and $S^1\times D^2 $, where M is a fiber bundle over $S^1$, with fiber $F$ such that its boundary is the knot $k$. ...
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Local Flatness and Knots

Peter Cromwell's `Knots and Links' has a definition for a locally flat knot that I'm struggling to understand. It's more the terminology used rather than the concept (I hope) but I can't find a ...
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Is a knot shadow always compatible with the trivial knot?

Define a knot shadow as a projection of a knot that does not indicate over- and under-crossings. So, if there are $c$ crossings, there are $2^c$ possible over/under assignments, and so that many ...
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Stereographic projection for a link/knot

I've been trying to understand the topological "link" between algebraic varieties and their associated knots/links and to this end I've been reading F. Kirwan's book, "Complex algebraic Curves". The ...
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How to find the braids that when closed make the $6_1$ knot.

I have the $6_1$ knot and my question is how can I easily find the braids that when closed make this knot, what's the easiest way in general for any knot.
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Correctness of the size of an planar integer lattice unknot

A planar integer lattice unknot is a polygon drawn over a two dimensional integer lattice. Here is an example: Given a number $N$, a planar unknot is not always possible. For example, a planar ...
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Actual Definition of the term: Hopf Band?

Sorry if this is too trivial: I need an actual working definition of the term: Hopf band. I see references to it in many searches, but never an actual precise definition. All I know so far is that ...
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Completeness of moves for polygonal knots

I am going through the paper, MINIMAL KNOTTING NUMBERS, by MANN et. al. On page six of the paper, they defined following moves for polygonal knots. Parallel moves Triangular moves I understand ...
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Why is surgery along a framed link well defined?

Let $L=L_1 \cup L_2 \cup \cdots \cup L_n $ be a framed link in $ S^3 $. I want to perform the surgery along $L$ to get a new manifold $M$. By definition, to perform this surgery, I must perform the ...
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52 views

More knots as crossing number increases?

Reading knot tables, it seems that as $n$ increases, more prime knots have crossing number $n$. Is this a proven fact? More precisely, If $k(n)$ is the number of knots with crossing number $n$, is ...
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Cohomology-Homology bilinear form of Seifert surfaces

Let $C_\ast$ be any chain complex of $R$-modules. Then for any $k\in\mathbb{Z}$ we obtain a $R$-bilinear map $$\langle-,-\rangle:H^k\!C_\ast\times H_kC_\ast\longrightarrow R, ...
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1answer
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Proof that knot genus is a knot invariant

I have a proof of the the following fact concerning knot genus, but I'm not sure that it's correct. If knot $J$ is isotopic to another knot $K$ then $J$ and $K$ have the same genus. Proof. Let ...
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Knot theory: Genus of a surface

Use Euler characteristic to determine the genus of the surface in Figure 4.24 in picture below. I am stuck with this question 4.10 from Colin Adams, the Knot Book.
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Crossing bound implies Reidemeister move bound?

In 1998 Galatalo established an upper bound on the number of Reidemeister moves needed to convert a diagram $D$ of the unknot into a trivial loop diagram. The upper bound is a function of $n$, the ...
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Seifert surfaces in Riemannian manifolds?

Does there exist an equivalent to Seifert surfaces for other Riemannian manifolds than $\mathbb{R}^3$? More precisely: Let $M$ be a simply-connected Riemannian manifolds and $K \subset M$ a (tame) ...
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Knot theory: Braids

Show by using a picture, that the two braids $\sigma_{i} \sigma_{i+1} \sigma_{i}$ and $\sigma_{i+1} \sigma_{i} \sigma_{i+1}$ are equivalent. This is 5.26 in knot book by Colin Adams. Need some ...
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the knot surgery - from a $6^3_2$ knot to a $3_1$ trefoil knot

It is intuitive that one can simply doing a cut-gluing surgery to make a $6^3_2$ to a $3_1$ trefoil knot: e.g. from to All one needs to do it to cut the three intersections at the angle of ...
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The math notation of this links? (connect sum of Hopf links)

We know the Hopf link owns the name of $2^2_1$ for Alexander–Briggs notations. (And there is another two component links is $4^2_1$.) I learned that "$4^3_1$ is not usually written as any three ...