For questions on knot theory, the study of mathematical knots

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34 views

Is there an invariant that completely classify all knots? [on hold]

Is there a set of invariants which completely classify all knots? And such that every object of this set represents a knot?
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1answer
29 views

A simple question about Jones polynomial

I understand the proof that the Jones polynomial, if its unique, is invariant under reidemeister moves, however how do we show that two inequivalent ways of computing it using the three rules and ...
3
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1answer
51 views

Current research on inverse knot equivalence?

What is the current status of the open problem in knot theory 'When is a knot equivalent to its inverse?' Additionally, I would like to know what work has been done on this problem (I cannot find ...
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1answer
26 views

When a 3-strand pretzel link is a knot?

I'm working on an article about the 3-strand pretzel links and I came across this remark : $P(p_1,p_2,p_3)$ is a knot if and only if none of two $p_i'$s are even. So I thaught that the the number of ...
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1answer
24 views

invariants of knots that are invariants under band move.

I am asking whether there are known knot invariants which are invariants under band move. Note that band move operation is similar to a connected sum of two knots except that the projections of two ...
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1answer
20 views

What exactly are strands in a quandle coloring?

From this slide on page 25, the system of equations: $$a = b \triangleleft (a \triangleleft b)$$ $$b = (a \triangleleft b) \triangleleft (b \triangleleft (a \triangleleft b))$$ Reduces down to: ...
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2answers
55 views

Fundamental group of $\mathbb{R}^3$ minus trefoil knot

Let $ \ T \subset \mathbb{R}^3 \ $ be the trefoil knot. A picture is given below. I need a hint on how to calculate the fundamental group of $ \ X = \mathbb{R}^3 \setminus T \ $ using Seifert-van ...
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0answers
54 views

QFT for mathematicians

I have a graduate degree in mathematics, I want to learn enough QFT to understand whats going on in Wittens paper about QFT and the Jones polynomial. So I need some QFT and maybe Chern-Simons theory. ...
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0answers
14 views

Computing the khovanov polynomial of the trefoil knot

I am reading the following article about khovanov polynomials of bar nathan: http://arxiv.org/abs/math/0201043 So far i have succeeded computing these polynomials for the hopf link, trivial link and ...
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0answers
37 views

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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0answers
30 views

Khovanov polynomial twisted unknot, trouble with factor spaces. [closed]

I am reading bar nathans paper about khovanov polynomials and am having a lot of trouble constructing the factor spaces. So suppose V is a vector space with basis ${v_{+},v_{-}}$ and of degree $\pm ...
3
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1answer
26 views

The isotopy type of the $Pretzel$ three strand knot $ P(a,b,c) $

I'm trying to prove that the knot $8_{20}$ is quasi-alternating. I thought of using the fact that this knot is actualy aquivalent to the $Pretzel$ knot $P(3,-3,2)$. There's this famous theorem that ...
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0answers
35 views

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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0answers
70 views

Knot theory and homology

What is the best way to learn about homology in knot theory? I am looking for a introductory book or resource, I dont know any homology, would I need to read a book about this first? If so, which?
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24 views

Is this game explained with knot theory or with homotopy theory? Or both?

The question is stated here. Obviously there exists an homotopy from the twist to the 'normal' circle because we are in $\mathbb{R}^3$, but I don't think there is always a solution to the game because ...
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1answer
28 views

Understanding the Kauffman bracket

"It follows easily from the bracket skein relation that a closed curve must count for a factor $\delta = −A^2 − A^{−2}$" Given the skein relation $< \times>=A<\, )(\, > + A^{-1}< \, ...
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1answer
97 views

Potential proof for the Slice-Ribbon conjecture (may be wrong).

Let $f:(D^2,S^1)\to(D^4,S^3)$ be a smooth embedding (so called a slice disk), and we set $M:=f(D^2)$. Then, is the restriction map $C^{\infty}(D^4)\to C^{\infty}(M)$ open map with relative to $C^2$ ...
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0answers
19 views

Ambient isotopy taking a polygon knot to another polygon knot

Let me define that a polygon knot means a knot K of which all point belongs to some line segment which is a subset of K. Let me ask if knot theory has some proof for that: For all pair ( K1, K2 ) ...
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2answers
55 views

Knots from the boundary of Möbius strips

A Möbius strip with one half twist has the unknot as its boundary. One with two half twists has a link of two unknots. One with three half twists has the trefoil knot as its boundary. Years ago, I ...
2
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1answer
33 views

figure-8 knot complement

The figure-8 knot seen as a 2-bridge knot with two maxima and two minima of the height function, has a complement in $S^3$ with one 0-handle,two 1-handles, two 2-handles and a 3-handle which cancels ...
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2answers
31 views

Are the generators of the braid group conjugates?

In his classic paper on Hecke algebra representations of braid groups from 1987, Vaughan Jones makes the claim that "the various generators $\sigma_i$ are all conjugate." How does one see this? I ...
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14 views

Alexander's theorem

Alexander's theorem states that any link can be realized as the closure of a braid. The hopf link for example is given by the closure of $\sigma_1^2$, and the trefoil knot is the closure of ...
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0answers
49 views

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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1answer
46 views

Can we embed $(p,q)$-torus knot in the torus without intersecting two disjoint non-trivial elements in homlogy

Let $T$ be a standard torus in 3-space. Let $[a]$ and $[b]$ be disjoint two closed paths in $T$ each of which is homotopic to a non-trivial element in homomlgy. In other words any of $[a]$ and $[b]$ ...
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1answer
30 views

On applications of Alexander's Theorem

I would like to know a bit about applications of the Alexander Theorem from Knot and Braid Theory. I would be very interested in learning about possible applications for the description of everyday ...
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2answers
18 views

About trivial torus-knot $T(p,q)$ such that $p$ or $q$ is one

Let $T(p,q)$ be a torus-knot where $p$ and $q$ are coprime. I am asking about the well known fact Which says if $p$ or $q$ is one, then $T(p,q)$ is trivial. If one of $p$ or $q$ is one, then we obtain ...
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1answer
29 views

Evaluating Colored Jones Polynomial of a trefoil knot

Following arXiv:1211.6075v1 I want to calculate colored Jones Polynomial for trefoil knot. I have the formulas: $ J_{\oplus R_i} = \sum_i J_{R_i} (K,q)$ $J_{R} (K^n, q) = J_{R^{\otimes n}} (K,q)$ I ...
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0answers
17 views

When connected sum of torus-links is slice?

Let $T(m,n)$ be a torus-link where $m$ and $n$ are any real numbers. My question is when connected sum of torus-links is slice?
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1answer
24 views

Obtaining a braid from a knot/link (Alexander's theorem)

I am following the algorithm for obtaining a braid from a link as explained on pages 23-24 here: https://math.berkeley.edu/~vfr/jonesakl.pdf. If I let the axis run straight through the center region ...
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1answer
20 views

What is the relationship between the Euler number of Seifert surface of a link and its linking number

Can I know whether the link is unlinked indeed (split Union of classical knots) from the Euler number of Seifert surface.
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25 views

How to prove a trefoil is not an unknot not using the theorem of Reidemeister moves

Let me ask how to prove that the trefoil class is not a sub class of the unknot class, not using the theorem of Reidemeister moves. The reason to exclude the theorem of Reidemeister moves is I have ...
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2answers
40 views

How to prove that two linked circles in $\mathbb{R^3}$ are not ambient isotopic to two circles showing no crossing

Let $C_1$ and $C_2$ be a pair of linked circles in $\mathbb{R^3}$ showing exactly two crossings. I want to know how to prove that no ambient isotopy takes them so that they will show no crossing, not ...
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2answers
77 views

Prerequisites for Kirby Calculus?

I've looked around, but I haven't found anything in particular on Google or here, so I figure I'd ask. What are some solid prerequisites to be able to tackle Kirby Calculus? I have a solid ...
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0answers
64 views

Exterior of a torus knot is Seifert-fibered.

I am trying to show that if $K\subset S^3$ is a $(p,q)$ torus knot, then the knot exterior $X_K=S^3\setminus N(K)$ is Seifert-fibered space, where $N(K)$ is a tubular neighborhood of $K$ in $S^3$. ...
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2answers
53 views

the evaluation of the Jones polynomial of an alternating link at $ t= -1 $.

I've been looking at some graph polynomials and I found a very nice relation between the famous Tutte polynomial of graphs and the no less famous Jones polynomial of links. Using this relation I was ...
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1answer
46 views

Knotted cross-sections of unknotted spheres

Livingston's notes on concordance mention "embeddings of $S^2$ into $\mathbb{R}^4$ which are unknotted, but have non-trivial knots as cross-sections. There are other such unknotted two spheres with ...
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1answer
24 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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0answers
78 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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0answers
28 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
30 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
31 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
9 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
24 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
28 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 ...
3
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1answer
92 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
80 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
18 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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22 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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1answer
83 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
30 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 ...