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Questions tagged [winding-number]

For questions about winding numbers. The winding number of a continuous curve counts how many times it "loops" around a given point.

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Example where rotation number of Legendrian knot depends on choice of Seifert surface

In Surgery on Contact Manifolds and Stein Surfaces there is the following exercise [below $Y$ is a 3-manifold, and $e(\xi)$ is the Euler class of $\xi$]: Exercise 4.2.6. Find a contact structure $(Y,\...
Hrhm's user avatar
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existance of a path that doesn't tuch an isotopy

Let $f_t :S^1 \to \mathbb{R}^2$ be an isotopy i.e. the function $F:[0,1]×S^1 \to \mathbb{R}^2$ defined by $F(t,x)= f_t(x)$ is continous, and each $f_t$ is homeomorphism onto it's image. let $\alpha \...
RT1's user avatar
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Simple Connectivity on the Complex Plane

There is a ton of equivalent definitions of a simply connected set on the plane, reading Ullrich's Complex Made Simple I have some of these equivalences, my question is about (iii) and (iv) ...
underfilho's user avatar
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Question about the logarithm function in the Argument Principle

I have found the proofs of the argument principle from Ponnusamy and Silverman's Complex variables with applications and Brown and Churchill's Complex variables. But I am not sure how to make sense of ...
nomadicmathematician's user avatar
3 votes
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Winding numbers in QCD and winding numbers in complex analysis. Is there a relation through a differential geometric generalization?

I have a background in theoretical physics and the first time I came across winding numbers was in the context of the vacuum of QCD. By the way physicists treat this topic, I thought it had little to ...
Carlos Bouthelier Madre's user avatar
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9 views

Possibility of superposition of nonpositive curvature curves produces positive turning

Suppose a finite set of plane curves parameterized by $t$: $p_i(t)=(x_i(t),y_i(t)), i=1...N$, satisfy boundedness: $|p_i(t)|\leq 1\ \ \forall i$ smoothness: $p_i(t)$ differentiable to any order, $\...
George C's user avatar
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Differential Topology and Winding number

I want to solve ex.1 from pag. 87 "Differential Topology" of Vietor Guillemin and Alan Pollaek. This is the first step of a sequence of exercises that should lead to the proof of the Jordan ...
Giovanni Barbarani's user avatar
1 vote
1 answer
73 views

Equivalence between two definitions of winding number

I've noticed that the definition of a winding number is rather different in Stewart & Tall's Complex Analysis than in Ahlfors' Complex Analysis: Let $\gamma:[a,b]\to\mathbb{C}$ be an arc and let $...
Sam's user avatar
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Is There a Conceptual Connection Between the 3D Winding Number and Ray Casting Algorithms?

The 3D winding number provides a numerical answer to whether a point is inside or outside a closed surface, with its definition arising from surface integration. In my recent journey through ...
K.R.Park's user avatar
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Two non closed paths homotopic relative to the endpoints have the same winding number?

Let $\gamma:[0,1]\rightarrow {C}$ be a path. The winding number $\omega(\gamma)$ is defined as $\omega(\gamma)=\frac{\theta(1)-\theta(0)}{2\pi}$. I have read that two paths homotopic to the endpoints ...
bananabob's user avatar
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Let $f(z)=\exp(z)\cos(z)$ prove $f(z)=a$ has infinite solutions outside any circunference.

I've been trying to solve this problem for a couple of days but i can´t find a convincent solution. The problem says: Let $f(z)=\exp(z)\cos(z)$. Prove for all $a \in \mathbb{C}$ except maybe one only ...
Bktr's user avatar
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For an arbitrary continuous curve $\gamma:[0,1]\to\mathbb R^n$, does the Riemann-Stieltjes integral $\int_0^1\gamma(t)\wedge d\gamma(t)$ exist?

We consider sums of the form $$\sum_{i=1}^m\gamma(t_i^*)\wedge\Big(\gamma(t_i)-\gamma(t_{i-1})\Big),$$ where $\gamma:[0,1]\to\mathbb R^n$ is a continuous function, and $$0=t_0\leq t_1^*\leq t_1\leq ...
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Getting a point in the interior of a polygon without relying on winding order?

I am given an arbitrary set of points embedded in 3D. The points are guaranteed to be ordered such that their order yields a simple closed polygon, but there is no information about whether they wind ...
Makogan's user avatar
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Soft question - Index theory in nonlinear dynamics vs Complex analysis

The video https://www.youtube.com/watch?v=wZvFKcQ_3Rc&t=8s mentioned something called the Index Theory. I can't find it on wikipedia. Where could I find more about the theory? Here index is just ...
HIH's user avatar
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Winding number of composition equaling zero implies existence of complex logarithm

I have an open and connected set $V$ that is also simply connected. Let $a_1, \ldots, a_n \in V$ and set $U = V \setminus \{a_1, \ldots, a_n\}$. Let $B_{R_i}(a_i)$ be disjoint balls around each $a_i$ ...
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Geometric characteristics or combinatorics analog for homotopy groups of spheres other than $\pi_n(S^n)$

The fact that $\pi_1(S^1) = \mathbb Z$ is geometrically intuitive. It is linked to a wealth of concepts and results, notably the winding number, residue theorem on $\mathbb C$, etc. There is a ...
Jean-Armand Moroni's user avatar
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Winding number of points on a path

I know that for $|z| = 1$ circular path with counter-clockwise rotation, the winding number of any point within the circle is 1(or more if the path is circulated more than 1 time. and also negative if ...
Maruf Hossain's user avatar
28 votes
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Homology in a picture? (Is this picture just metaphorical, or a rigorous example that can be formalized?)

A post-doc colleague showed me this picture and said: going from the diagram No.2 to No.3 and to No.4 is taking the homology. I did not quite understand this comment. For me, if I take simplicial ...
gwynneth-m.sc.'s user avatar
2 votes
0 answers
83 views

Winding Numbers and Simply Connected Sets [closed]

Suppose $D$ if a path connected open subset of $\mathbb C$ so that the winding number of any $x$ in $\mathbb C -D$ with respect to any loop lying in $D$ is 0. Why does this imply that every loop lying ...
Drooga's user avatar
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Winding number of external point

Let $γ$ be a closed (C1-)curve whose image is contained in ${z: |z| < R}$ for some $R > 0$. Show that for any $z$ with $|z| > R$ we have $\operatorname{Ind}(γ,z) = 0$. This case is clear to ...
Crash Bandicoot's user avatar
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Winding number of two curves defined on the same interval

This question is about an observation regarding winding numbers. Let $\gamma_1$ and $\gamma_2$ be two closed curves defined on the same interval $[a,b]$ onto $\mathbb{R}^2\setminus \{(0,0)\}$, each ...
Michael Wang-Wakamatsu's user avatar
2 votes
2 answers
144 views

calculating winding number around zero [duplicate]

let $m, n \in \mathbb{Z}$ be fixed and let $ 0 < r \neq 1$, determine the winding number of the closed curve $\gamma(t) : = e^{imt} + re^{int}$ $\gamma: [0, 2\pi] \to \mathbb{C} \backslash 0$ ...
galaxy--'s user avatar
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Compute an oriented angle in an abstract plane.

I am coding something a bit esoteric. I have a stream of points which are guaranteed to be planar, however the points are embedded in some high dimensional space (e.g. 3D). I need to define a notion ...
Makogan's user avatar
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Why is the winding number of a holomorphic function the number of solutions to $f(z)=f(z_0)$?

In class the following situation was presented: suppose $f:\Omega\to \mathbb{C}$ is holomporphic. Let $z_0\in\Omega$ and suppose $f'(z_0)=0$ but $f''(z_0)\not=0$.Now consider the circle $C$ of radius $...
Stijn D'hondt's user avatar
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Mapping with infinite values - Winding number and complex analysis

Is there a continuous closed curve $\gamma$ in $\mathbb{C}$ with the property that the mapping $$ \nu_{\gamma}: \mathbb{C} \backslash \operatorname{im}(\gamma) \rightarrow \mathbb{Z} $$ takes ...
calculatormathematical's user avatar
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Does the path $t \mapsto \zeta(\frac 12 + it)$ eventually stay clockwise?

When we plot the path $\gamma(t):= \zeta (\frac 12+it)$ starting at $t=0$ and letting $t$ increase, we get the following beautiful image (courtesy of 3b1b's Riemann zeta function video's thumbnail), ...
D.R.'s user avatar
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Find a closed path so that the number of zeroes is equal to the winding number

Let $D= \{ z\in \mathbb{C}: |z|<1\}$ and $T= \{ z\in \mathbb{C} : |z|=1\}$. Let $f: \Omega \rightarrow \mathbb{C} $ be analytic on an open set $\Omega \supset D \cup T$, and suppose that $f(z)\ne0$,...
Korn's user avatar
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5 votes
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What type of Mathematics, if any, is this? (On curiosities associated with a logo.)

I'm not sure whether I have articulated my curiosity well enough here. Please, therefore, bear with me if I need to edit the question, and please forgive me if this is otherwise a nonsense question ...
Shaun's user avatar
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1 vote
1 answer
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Is the winding number of a null-homologous cycle outside an open ball zero?

Let $U\subset{\mathbb{C}}$ be open and $c=n_1•c_1+…+n_k•c_n$ (formal sum with $c_i:[0,1] \to U$ continuous, $n_i \in \mathbb{Z}$) be a null-homologous cycle (closed 1-chain with $Ind_c(z)=0 \text{ for ...
LucesAim12's user avatar
2 votes
1 answer
217 views

Winding number of an ellipse $\gamma$ at $0$ using $\omega(\gamma,0)=\frac{1}{2i\pi}\oint_{\gamma}\frac{1}{z}dz$?

In this exercise, we are supposed to firstly find a path that parametrizes the following ellipse: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ for $a,b \in \mathbb{R}$ $\textit{I have found the following path: ...
cnymfais's user avatar
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3 answers
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How to show that the winding number must be $0$ or $\pm$ $1$ in this case?

Prerequisite: $K: [0,1] → \mathbb{C}$ piecewise, continuously differentiable, closed path that meets the non-positive real axis at only at one point $K^{-1} (\{x|x \le 0\})= \{ t_0 \}$ with $0 < ...
galaxycrash's user avatar
6 votes
1 answer
311 views

Regarding the winding number

My main question is about part B, but I would also be grateful if you can tell me what you think about part A. Define a smooth vector field $X$ on $S^1$ as follows: $X(x,y)=(-y,x)$. For a smooth map $...
Lam18373's user avatar
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A question about the formula for winding numbers in SMT 2021 Power Round

I was looking at the Power Round of Stanford Math Tournament and I came across this formula for a winding number of a curve: $$w(\gamma)=\frac{1}{2πi}\int_0^{2π}\frac{\gamma’(s)}{\gamma(s)} \mathrm {...
CrabSis's user avatar
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2 votes
0 answers
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Maps that preserve winding numbers

Update: now crossposted to MathOverflow: https://mathoverflow.net/questions/419705/maps-that-preserve-winding-numbers I am looking for a characterisation of the continuous maps on some subset of $A\...
Manuel Eberl's user avatar
3 votes
1 answer
269 views

Closed rectifiable curve with arbitary winding number.

This is an exercise from Conway's Functions of One Complex Variable(page 83, exercise 2). Give an example of a closed rectifiable curve $\gamma$ in $\mathbb C$ s.t. for any integer $k$ there is a ...
William Leynoid's user avatar
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Representations of $\mathfrak{u}(1)$ on $\mathbb{C}$ and $\bar{\mathbb{C}}$ with winding number $k\neq0$ are not isomorphic.

In the text I'm reading, $U(1)$ has a representation on $\mathbb{C}$ with winding number $k$ given by: $$z\mapsto z^k$$ I'm supposed to show that the representations of $\mathfrak{u}(1)$ with winding ...
Chris's user avatar
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3 votes
1 answer
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"Winding number" of the map $g: T^{2n+1} \to U(N)$

For $n\in \mathbb Z_+$, let $T^{2n+1}$ be the torus and $U(N)$ be the unitary group, where $N$ is sufficiently large. A physics paper on topological insulators claims the following: For a map $g: T^{...
Laplacian's user avatar
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How do I compute the local degree of the following map?

I have the following porblem. Let $f:\mathbb{R}^2\rightarrow \mathbb{R}^2$ be a linear mapping, represented by a $2\times 2$ matrix. Moreover let the determinant be nonzero. Show that the local ...
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2 votes
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How do I show that the winding number of this closed continuous path is zero?

I have the following problem: Show that if $\gamma:[a,b]\rightarrow U$ is a closed continuous path, $p\in \mathbb{R}^2$ and $U\subset \mathbb{R}^2\setminus \{p\}$ is an open subset on which there is ...
user1294729's user avatar
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If $ |\phi_1| <| \phi_2| $ and $\phi = \phi_1 + \phi_2 $ then $\mathrm {Ind} _{\phi} (0) = \mathrm {Ind} _{\phi_2} (0)$

Let $ \phi1, \phi2: [0,1] \to \mathbb {C} $ be Paths, such that $ |\phi_1 (t) | <| \phi_2 (t) | $ for all $ t \in [0,1] $. If $ 0 \leq t \leq 1 $, we set $\phi (t) = \phi_1 (t) + \phi_2 (t)$. Prove ...
Made's user avatar
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1 vote
1 answer
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Small perturbations of a loop in $\mathbb C$ does not change its winding number

From Visual Complex Fuctions by Elias Wegert, Lemma 2.7.19, it says that Let $\gamma_0 : [0,1] \to \mathbb C - \{0\}$ be a loop, i.e. a continous function with same endpoints. Denote by $d$ the ...
Squirrel-Power's user avatar
2 votes
2 answers
295 views

How can I prove that this defintion of a winding number is valid?

I have read a definition of a winding number on wikipedia and it involves finding the continuous polar parametrization of the curve, but then the question arises, why does such a parametriation always ...
Юрій Ярош's user avatar
1 vote
0 answers
59 views

Does injective continuous / holomorphic functions preserve winding number?

I am trying to find if the winding number is invariant to some transformations. I already know that it is invariant respect to translation, rotation and scaling. Also, it is not invariant respect to ...
Jose Pérez Cano's user avatar
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0 answers
56 views

Can I compute contour orientation without using polygon area sign?

Most of the times, I determine contour orientation generating 2D points and computing the closed polygon area. Depending on the area value sign I can understand if the contour is oriented clockwise or ...
abenci's user avatar
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1 answer
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How do I prove this statement about winding numbers and continuous maps?

I have the following problem: Let $D$ be a disk with boundary circle $C$ and let $f:D\rightarrow \mathbb{R}^2$ be a continuous map. Suppose $P\in \mathbb{R}^2\setminus f(C)$ and the winding number of ...
user123234's user avatar
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1 vote
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Winding Number of Linear Map

Let $f: \mathbb{R}^2 \to \mathbb{R}^2$ be a linear map, represented by a matrix $A \in \mathbb{R}^{2 \times 2}$, with non-zero determinant. Further, define the path $\gamma:[0;1]\to \mathbb{R}^2$, $$\...
Vivian's user avatar
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$W(\gamma +v , p+ v) = W(\gamma, p)$ why is this true?

I don't understand why this statement in lecture is true : let $\gamma : [a,b] \rightarrow \mathbb{R}^2 \setminus \left \{ p\right \} $ a continuous path, and v any vector in the plane. Moreover, ...
Bünzli Refinej's user avatar
1 vote
0 answers
78 views

Embedding of graphs in topological spaces

I am studying a problem concerned with the embedding of a graph $G=(V,E)$ in spaces that are not simply connected, e.g. $(\mathbb{C}\setminus\{0\})\times\mathbb{R}$. My understanding is that an ...
Tanatofobico's user avatar
1 vote
1 answer
359 views

usage of Rouche's theorem for winding number in proof of Residue theorem

A doubt from proof of residue theorem from Tao's notes. notes 4 Theorem 21 (Residue theorem) Let $U\subset \mathbb{C}$ be a simply connected open set, and let $f: U\setminus S\rightarrow \mathbb{C}$ ...
Vinay Deshpande's user avatar
2 votes
0 answers
44 views

A question about such integrations relating to winding numbers

I am wondering, since $$ \frac{1}{2 \pi i} \int_{\gamma} \frac{f^{\prime}(z)}{f(z)} d z $$ can offer information about winding numbers by $$ \frac{1}{2 \pi i} \int_{\gamma} \frac{f^{\prime}(z)}{f(z)} ...
Zizheng Yang's user avatar