# 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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### 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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### 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 ...
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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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### 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 ...
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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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 ...
1 vote
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### Why is $n(F,D)$ odd, when $F$ is odd?

This is a question from Do Carmo's Differential Forms and Applications (question 8, chapter 2). Actually, this question was made and answered here. The problem is: The answer redirects the OP to here (...
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### Winding number is locally-constant for general curves (not $C^1$) using variation of argument definition

I've been searching for this proof, here and on the web too, but it seems like the answer is harder to find than expected. There are many similar questions about this but they all (implicitly or ...
### Calculate the integral $\int_{0}^{1} \frac{1}{z_0+t(z_1-z_0)-z} \,dt$.
Let $z, z_0, z_1$ be three different complex numbers. I am trying to calculate the integral $\int_{0}^{1} \frac{1}{z_0+t(z_1-z_0)-z} \,dt$. Is there a closed form? Any hints? Thank you.
I am trying to prove that the winding number $n(p\circ\gamma_r,0)$ is $k$, when $p(z)=a\prod_{j=1}^k(z-z_j)$ is a complex polynomial of degree $k$ and $\gamma_r:[0,2\pi] \rightarrow \mathbb{C}, \... 0 votes 0 answers 174 views ### How winding number in a boundary point of a curve is defined? We know that, the winding number counts the number of rounds of a curve around a point. It is a positive integer for positive oriented curves and a negative integer for curves with negative ... 0 votes 0 answers 141 views ### Is this set a star domain? Definition: A set S in$\mathbb{C}$is called a star domain if there exists an$a \in S$such that for all$x \in S$, the line segment from a to x is in S. And the point$a \in S$is called a star ... 0 votes 1 answer 52 views ### if$f\in C(\mathbb{\overline{D}})$and$\forall z\in\partial\mathbb{D}.f(z)=z$then$\exists z\in\mathbb{D}.f(z)=0$if$f\in C(\mathbb{\overline{D}})$and$\forall z\in\partial\mathbb{D}.f(z)=z$then$\exists z\in\mathbb{D}.f(z)=0\$ f is not analytical only Continuous . solve with complex analysis tools I thought ... 