False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

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The winding number depends only on the trace of the closed contour

Statement

False claim. If two closed complex contours have the same trace, then they have the same winding number about every point off that trace.

Facts & Assumptions

Given: The contours $γ_{1} (t) = \exp (i t)$ and $γ_{2} (t) = \exp (2 i t)$ on $[0, 2 π]$ .

[L1]

For $a \in C$ , $r > 0$ and $k \in Z$ , the contour $γ_{k} (t) = a + r \exp (i k t)$ on $[0, 2 π]$ is a closed complex contour with $n (γ_{k}, z) = k$ for $∣ z - a ∣ < r$ and $n (γ_{k}, z) = 0$ for $∣ z - a ∣ > r$ ; for $k \neq 0$ its trace is ${z : ∣ z - a ∣ = r}$ (A circle traversed $k$ times has winding number $k$ inside and $0$ outside).

[L2]

For a closed complex contour $γ$ and $p$ off its trace, $n (γ, p) = (2 π i)^{- 1} \int_{γ} d z / (z - p)$ , a quantity defined from the parametrised contour (The winding number of a closed contour about a point off its trace).

[L3]

A complex contour is a rectifiable path together with its parameter interval and its parametrisation; its trace is only the image set (Rectifiable complex contours, reversal, concatenation, closedness, and orientation).

Refutation

technique · direct

1.1

By [L1] with $a = 0$ , $r = 1$ and $k = 1$ , the contour $γ_{1}$ is closed with trace ${∣ z ∣ = 1}$ and $n (γ_{1}, 0) = 1$ .

givenL1

1.2

By [L1] with $a = 0$ , $r = 1$ and $k = 2$ , the contour $γ_{2}$ is closed with trace ${∣ z ∣ = 1}$ and $n (γ_{2}, 0) = 2$ .

givenL1

2.1

The two contours have the same trace, and $0$ lies off it, yet $1 \neq 2$ ; so the claim is false.

step 1.1step 1.2

3.1

Nothing here is anomalous: by [L2] the index is computed from an integral over the parametrised contour, and by [L3] the trace forgets the parametrisation, which is what records how many times the circle is traversed. This is why The winding number of a closed contour about a point off its trace attaches the index to the map and not to the image set.

step 2.1L2L3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

37 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

L. V. Ahlfors, Complex Analysis, 3rd ed., Ch. 4 §2.1 (standard reference, not scraped)