For questions on knot theory, the study of mathematical knots

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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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29 views

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

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 ...
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1answer
389 views

Equivalence of knots: ambient isotopy vs. homeomorphism

I am looking into knot theory and have found two different definitions stating that two knots $K_1$ and $K_2$ are equivalent, namely the concept of an ambient isotopy: These two knots are ambient ...
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1answer
20 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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31 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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39 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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25 views

Readable references on gauge theory, knot theory and related

I have read the books Baez, Gauge theory knots and gravity and adams the knot book. When I try to follow the references there seems to be a big gap in readability and prerequisities required. Are ...
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21 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
25 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
85 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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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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31 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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50 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 ...
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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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30 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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1answer
43 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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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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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
28 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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1answer
22 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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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
19 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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39 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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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
76 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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59 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
51 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
45 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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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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75 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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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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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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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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28 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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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
27 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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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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88 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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63 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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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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56 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
81 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 ...
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2answers
142 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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67 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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178 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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95 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 ...