Fact-checked by Grok 2 weeks ago

Winding number

In mathematics, particularly in the fields of topology and complex analysis, the winding number of a closed curve around a point in the plane is an integer that quantifies the net number of times the curve encircles the point, with positive values indicating counterclockwise orientation and negative values indicating clockwise orientation.^[1] For a point not lying on the curve, this measure captures the topological linking between the curve and the point, serving as a homotopy invariant that remains unchanged under continuous deformations of the curve that do not pass through the point.^[2] Formally, for a smooth closed curve C parameterized by c(t) = (x(t), y(t)) for t \in [0, T] with c(0) = c(T), and a point p \notin C, the winding number W(p, C) can be computed as the total change in the polar angle subtended by the curve at p, divided by $2\pi:
W(p, C) = \frac{1}{2\pi} \int_0^T \frac{ -y(t) \dot{x}(t) + x(t) \dot{y}(t) }{ x(t)^2 + y(t)^2 } \, dt ,
where the coordinates are relative to p, and the result is always an integer.^[1] In the context of complex analysis, for a closed curve \gamma in the complex plane and a point z_0 \notin \gamma, the winding number n(\gamma, z_0) is equivalently given by the contour integral
n(\gamma, z_0) = \frac{1}{2\pi i} \int_\gamma \frac{dz}{z - z_0} ,
which counts the oriented revolutions of \gamma around z_0.^[3] Key properties of the winding number include its additivity under concatenation of curves, where W(p, C_1 \cdot C_2) = W(p, C_1) + W(p, C_2), and its role as a complete invariant for homotopy classes of curves in the punctured plane.^[4] For a simple closed curve, such as the boundary of a Jordan domain, the winding number is +1 or -1 for points in the bounded interior component (depending on orientation) and $0 for points in the unbounded exterior component, underpinning the Jordan curve theorem.^[4] In complex analysis, it features prominently in the argument principle, which equates the winding number of the image curve f(\gamma) around the origin to the number of zeros minus poles of an analytic function f inside \gamma, enabling proofs of theorems like the fundamental theorem of algebra.^[5] These attributes make the winding number a versatile tool in algebraic topology, geometric analysis, and computational geometry.^[6]

Introduction

Intuitive Description

The winding number of a closed curve in the plane around a given point quantifies the net number of complete loops the curve makes around that point, where counterclockwise encirclements count as positive and clockwise ones as negative. This integer value captures the overall "twisting" or encircling behavior of the curve relative to the point, providing a simple measure of its topological enclosure without requiring advanced mathematics. For instance, if the point lies outside the curve entirely, the winding number is zero, indicating no net encirclement.^[7] Consider basic examples to build intuition: a circular curve traced counterclockwise around the point yields a winding number of 1, as it completes one full positive loop. In contrast, tracing the same circle clockwise gives -1. A figure-eight shaped curve, which crosses itself and forms two opposing loops around a central point, results in a net winding number of 0, since the counterclockwise loop (+1) cancels the clockwise one (-1). For more complex self-intersecting curves, the winding number sums the contributions from each relevant loop, yielding the algebraic total rather than an absolute count.^[8]^[9] An everyday analogy helps visualize this: imagine walking your dog on a leash through a park with a tall tree at its center. If the dog darts around the tree while you continue your path, the number of times the leash wraps around the trunk—netting positive for one direction and negative for the other—mirrors the winding number, as the leash's total revolutions remain invariant regardless of minor path variations, as long as the tree stays enclosed similarly.^[10] Diagrams often illustrate this by plotting the angle from the fixed point to positions along the curve as it travels; the total angular sweep, divided by $2\pi, reveals the winding number through the number of full rotations accumulated. This angle-tracking approach underscores the rotational intuition behind the concept.^[11]

Historical Overview

The concept of the winding number originated in 19th-century complex analysis through Augustin-Louis Cauchy's foundational work on contour integrals. In his 1825 publication of the integral theorem, Cauchy established that the integral of a holomorphic function over a closed contour depends on the singularities enclosed, implicitly incorporating ideas of how paths encircle points in the complex plane.^[12] This was further developed in Cauchy's argument principle, which quantifies the number of zeros and poles inside a contour via the total change in argument along the path, a measure directly analogous to the winding number as the net encirclements divided by $2\pi.^[13] In the early 20th century, the winding number gained more explicit formalization through refinements in complex analysis. Édouard Goursat's 1900 proof of Cauchy's theorem, published without reliance on residues or continuity of the derivative, highlighted the theorem's dependence on path homotopy and the vanishing of integrals over contractible paths, concepts tied to zero winding numbers.^[14] This proof emphasized the topological independence of integrals from the specific contour shape, provided no singularities are enclosed, laying groundwork for viewing winding as a homotopy invariant. In parallel, the topological perspective emerged with Henri Poincaré's introduction of the fundamental group in the early 1900s, where the winding number represents the generator of π₁ of the punctured plane or circle.^[15] The adoption of the winding number in topology accelerated in the 1930s with Heinz Hopf's contributions to degree theory and fibrations. Hopf's 1931 work on the topological invariant for maps between spheres introduced the Hopf invariant, linking winding-like measures to homotopy classes and fundamental groups, particularly for circles and higher-dimensional analogs.^[16] Post-World War II expansions in algebraic topology further integrated the winding number into homotopy theory through works like Witold Hurewicz's development of higher homotopy groups (1935) and the Hurewicz theorem (1940s), which relate homotopy and homology groups in low dimensions, encompassing fundamental group invariants such as the winding number.^[17] A comprehensive survey of these evolutions across topology, geometry, and analysis appears in John Roe's 2015 monograph, which traces the winding number's role from classical theorems to modern invariants.^[18] Recent developments have extended discrete versions of winding numbers to computational geometry, particularly for handling noisy or imperfect data. The 2023 SIGGRAPH paper by Feng, Gillespie, and Crane introduces an algorithm for computing winding numbers on discrete surfaces with topological errors, enabling robust point-in-polygon tests and signed distance approximations in practical graphics applications.^[19]

Definitions and Formulations

Basic Formal Definition

The winding number of a closed curve \gamma: [0,1] \to \mathbb{R}^2 \setminus \{p\} around a point p \notin \gamma([0,1]) is defined as

n(\gamma, p) = \frac{1}{2\pi} \int_{\gamma} d[\theta](/page/Theta),

where \theta denotes the angle that the vector from p to points on \gamma makes with a fixed reference direction, and the integral represents the total variation in \theta as one traverses \gamma. This measures the net number of revolutions the curve makes around p in the counterclockwise direction. By translating coordinates so that p maps to the origin $0 \in \mathbb{C}$, the definition admits an equivalent complex-analytic form

n(\gamma, 0) = \frac{1}{2\pi i} \int_{\gamma} \frac{dz}{z},

valid for \gamma parametrized as a closed path in \mathbb{C} \setminus \{0\}. The winding number is integer-valued: to see this, consider the continuous argument function \theta: [0,1] \to \mathbb{R} along \gamma such that \theta(t) is the angle of \gamma(t) - p, normalized so the map t \mapsto e^{i \theta(t)} traces a loop in S^1. Since \gamma is closed, \theta(1) \equiv \theta(0) \pmod{2\pi}, so the total change \Delta \theta = \theta(1) - \theta(0) is a multiple of $2\pi, yielding n(\gamma, p) = \Delta \theta / (2\pi) \in \mathbb{Z}.^[20] Key properties include additivity under concatenation of curves, n(\gamma_1 \cdot \gamma_2, p) = n(\gamma_1, p) + n(\gamma_2, p), and vanishing if \gamma is contractible in \mathbb{R}^2 \setminus \{p\} (hence n(\gamma, p) = 0 whenever p lies outside the region "enclosed" by \gamma in a homotopical sense).^[20] The winding number is undefined if p \in \gamma([0,1]), but extends continuously to zero in such limiting cases where the curve avoids p but approaches it without enclosing it. For example, consider the unit circle \gamma(t) = e^{2\pi i t} for t \in [0,1] around p = 0. Here, \theta(t) = 2\pi t, so \Delta \theta = 2\pi and n(\gamma, 0) = 1; the complex integral form confirms this as \frac{1}{2\pi i} \int_{\gamma} \frac{dz}{z} = 1 by direct parametrization.

In Complex Analysis

In complex analysis, the winding number provides a key tool for analyzing the behavior of meromorphic functions along closed contours. For a meromorphic function f and a closed contour \gamma in the complex plane such that a \notin f(\gamma), the winding number n(f \circ \gamma, a) of f \circ \gamma around the point a is defined by the contour integral

n(f \circ \gamma, a) = \frac{1}{2\pi i} \int_\gamma \frac{f'(z)}{f(z) - a} \, dz.

This formula arises from the change in the argument of f(z) - a as z traverses \gamma, divided by $2\pi, and it counts the net number of times f(\gamma) encircles a in the positive direction.^[3] The argument principle extends this concept to relate the winding number directly to the zeros and poles of f inside \gamma. Specifically, if f is meromorphic in a domain containing \gamma and its interior, with no zeros or poles on \gamma, then

\frac{1}{2\pi i} \int_\gamma \frac{f'(z)}{f(z)} \, dz = N_f - P_f,

where N_f is the number of zeros of f inside \gamma (counted with multiplicity) and P_f is the number of poles (also with multiplicity). Thus, the winding number n(f \circ \gamma, 0) equals the difference between the number of zeros and poles of f enclosed by \gamma. This principle is fundamental for locating zeros and poles without explicit solving.^[3] The winding number also underpins Cauchy's integral theorem in its generalized form. For a holomorphic function h in a simply connected domain, the integral \int_\gamma h(z) \, dz = 0 if \gamma is homologous to zero in the domain, meaning the winding number of \gamma around any singularity of h (though h has none) is zero. More broadly, if h is holomorphic inside and on \gamma with no singularities enclosed, the winding number condition ensures the integral vanishes, highlighting how the absence of encircled singularities implies path independence of integrals. Rouché's theorem leverages the winding number to compare the zero structures of two functions. If f and g are holomorphic inside and on a simple closed contour \gamma, and |g(z)| < |f(z)| for all z \in \gamma, then f and f + g have the same number of zeros inside \gamma (counted with multiplicity). This follows because the winding numbers n(f \circ \gamma, 0) and n((f + g) \circ \gamma, 0) coincide, as g/f maps \gamma into the unit disk, deforming the image without crossing zero.^[3]

In Algebraic Topology

In algebraic topology, the winding number of a closed oriented curve \gamma: S^1 \to \mathbb{R}^2 \setminus \{p\} around a point p is defined as the homotopy class of the associated map \tilde{\gamma}: S^1 \to S^1 obtained by normalizing the vector from p to points on the curve, i.e., \tilde{\gamma}(t) = \frac{\gamma(t) - p}{|\gamma(t) - p|}. This class lies in the set of homotopy classes [S^1, S^1], which is isomorphic to \mathbb{Z}, where the integer representative corresponds to the degree of \tilde{\gamma}.^[20]^[6] The fundamental group \pi_1(S^1) is isomorphic to \mathbb{Z}, generated by the class of the standard loop that traverses the circle once counterclockwise. For loops based at a point on S^1, the induced homomorphism on \pi_1 sends the generator to n times the generator of \mathbb{Z}, where n is the winding number, making it a complete invariant for homotopy classes of based loops in the circle. In the plane complement \mathbb{R}^2 \setminus \{p\}, which is homotopy equivalent to S^1, loops based away from the curve map similarly to integers via this isomorphism.^[20] The winding number is a homotopy invariant: if two loops \gamma_0 and \gamma_1 are homotopic relative to the basepoint in \mathbb{R}^2 \setminus \{p\}, then their normalized maps \tilde{\gamma_0} and \tilde{\gamma_1} are homotopic in [S^1](/page/S^1), preserving the degree and thus the winding number. This invariance follows from the continuity of the normalization map under homotopy and the fact that [S^1, S^1] \cong \mathbb{Z} classifies such maps up to homotopy.^[20]^[6] This concept generalizes to the degree of continuous maps f: S^n \to S^n for n \geq 1, where the degree is an integer invariant classifying homotopy classes [S^n, S^n] \cong \mathbb{Z}, with the case n=1 recovering the winding number as the induced action on \pi_1(S^1) \cong \mathbb{Z}. For n=1, the degree measures how many times the domain wraps around the codomain. An illustrative example is the projection of the trefoil knot (a (2,3)-torus knot) onto the plane: the immersed curve winds around a central point with winding number 2, reflecting its topological embedding properties.^[20]^[21]

In Differential Geometry

In differential geometry, the winding number of an oriented smooth closed curve \gamma: S^1 \to \mathbb{R}^2 around a point p \notin \gamma(S^1) is defined as

n(\gamma, p) = \frac{1}{2\pi} \int_\gamma d \arg(\gamma(t) - p),

where \arg is the argument function measuring the angle of the vector \gamma(t) - p with respect to a fixed axis, and d \arg denotes its exterior derivative as a differential 1-form on \mathbb{R}^2 \setminus \{p\}.^[22] This expression quantifies the net rotation of the direction from p to points on \gamma as the curve is traversed once, yielding an integer value that counts the algebraic number of loops around p.^[22] Equivalently, n(\gamma, p) is the topological degree of the normalized Gauss map g: S^1 \to S^1 given by g(t) = \frac{\gamma(t) - p}{\|\gamma(t) - p\|}, computed as the integral of the pullback of the normalized volume form on the target circle:

n(\gamma, p) = \int_{S^1} g^* \left( \frac{d\theta}{2\pi} \right),

where \theta is the standard angular coordinate on S^1.^[23] This formulation emphasizes the winding number as a de Rham cohomology class representative, capturing the homotopy class of \gamma relative to p in the punctured plane.^[23] For immersed smooth closed curves in the plane, the rotation index \nu(\gamma) provides a related geometric invariant, defined as the degree of the tangent indicatrix map \tau: S^1 \to S^1, \tau(t) = \frac{\gamma'(t)}{\|\gamma'(t)\|}, or equivalently,

\nu(\gamma) = \frac{1}{2\pi} \int_\gamma d\phi,

where \phi is the angle of the unit tangent vector \tau(t).^[24] The total signed curvature of \gamma satisfies \int_\gamma \kappa_g \, ds = 2\pi \nu(\gamma), linking the winding behavior of the tangent to the curve's global turning properties; for simple closed curves, \nu(\gamma) = \pm 1.^[24] This definition extends naturally to piecewise smooth curves, where the integral over each smooth arc is computed separately, and contributions from jumps in the tangent at finitely many vertices are included via the exterior angles, ensuring the winding number remains an integer as long as p lies off the curve.^[25] In contrast, purely smooth curves avoid such discrete adjustments, allowing direct evaluation via the continuous differential form without vertex terms.^[25] The value is homotopy invariant under deformations fixing p.^[24] A representative example is the orthogonal projection onto the xy-plane of a helical space curve \gamma(t) = (a \cos t, a \sin t, b t) for t \in [0, 2\pi n] with a > 0, b > 0, and integer n \geq 1, which yields the parametric curve (\cos t, \sin t) traversed n times, possessing winding number n around the origin.^[26]

Turning Number

The turning number of a plane curve \gamma is defined as the total rotation of its tangent vector as one traverses the curve, measuring the net change in the direction of the tangent divided by $2\pi.^[27] For a smooth curve parametrized by arc length s, this is given by \tau(\gamma) = \frac{1}{2\pi} \int_{\gamma} \kappa \, ds, where \kappa is the signed curvature, or equivalently, \tau(\gamma) = \frac{1}{2\pi} \Delta \theta, with \Delta \theta the total change in the tangent angle \theta.^[27] For a simple closed oriented plane curve, the Hopf Umlaufsatz theorem states that the turning number is \pm 1, with the sign determined by the orientation (positive for counterclockwise); this value is independent of the specific embedding of the curve, depending only on its topological type as a simple closed loop.^[27] Unlike the winding number, which quantifies how many times a closed curve encircles a specific point in the plane, the turning number is an intrinsic property of the curve itself, capturing the self-rotation of its tangent without reference to an external point; for a simple closed curve, the magnitudes coincide when the winding is computed around an interior point, but the concepts differ fundamentally in their geometric interpretation.^[27]^[28] Examples include a circle, which has turning number $1 (or -1 if oriented clockwise), reflecting one full rotation of the tangent; a straight line segment, with turning number $0 due to no net rotation; and the projection of a space curve onto the plane, where the turning number may differ from the original space curve's total curvature if the projection introduces apparent turns or straightenings.^[27] For polygonal curves, the turning number is computed as the sum of the exterior (turning) angles at the vertices divided by $2\pi; for a simple closed polygon, this sum is \pm 2\pi, yielding \tau = \pm 1.^[27]

Linking Numbers

Using Alexander duality, the first homology group of the complement of a knot or link in the 3-sphere S^3 provides a homology-theoretic framework that generalizes the winding number to higher dimensions, capturing how components wind around each other via linking numbers. For a single embedded circle K \subset S^3, Alexander duality implies that H_1(S^3 \setminus K; \mathbb{Z}) \cong \mathbb{Z}, generated by the class of a meridian loop around K.^[29]^[30] This infinite cyclic group encodes the basic winding structure: the image of a closed curve in the complement under the inclusion-induced map to H_1(S^3 \setminus K) yields an integer multiple of the generator, representing the winding number of that curve around K. For the unknot, this homology remains \mathbb{Z}, with the integer reflecting the trivial embedding's single meridional winding.^[29] For an oriented link L = K_1 \cup \cdots \cup K_\mu \subset S^3 with \mu \geq 2 components, Alexander duality yields H_1(S^3 \setminus L; \mathbb{Z}) \cong \mathbb{Z}^\mu, freely generated by the meridional classes [m_1], \dots, [m_\mu] of the components.^[29] The linking numbers between components are defined via the inclusion maps: the class [K_i] in H_1(S^3 \setminus K_j; \mathbb{Z}) \cong \mathbb{Z} (for i \neq j) is the integer \mathrm{lk}(K_i, K_j) times the generator [m_j], measuring how K_i winds around K_j.^[31] These pairwise linking numbers generalize the classical planar winding number to three dimensions, capturing mutual encirclements in the link homology. In higher dimensions, for a knotted sphere S^k \subset S^{n} with n > 3, Alexander duality extends this to H_{n-k-1}(S^n \setminus S^k; \mathbb{Z}), providing analogous invariants for multidimensional windings.^[29] These linking numbers can also be realized geometrically via Seifert surfaces: for distinct components K_i and K_j, \mathrm{lk}(K_i, K_j) equals the algebraic intersection number of K_i with any oriented Seifert surface bounded by K_j.^[31] This intersection perspective directly generalizes the planar winding number, where the "surface" is a disk and intersections count encirclements. For the Hopf link, consisting of two unknotted components interlocked once, the linking numbers are \pm 1 (depending on orientations), yielding the simplest nontrivial example.^[31] As a topological invariant derived from homology groups via Alexander duality, the linking number is preserved under ambient isotopy of the link in S^3, ensuring it distinguishes link types robustly. Note that the Alexander polynomial, a further invariant from the homology of the infinite cyclic cover, builds on this structure to detect more subtle knot properties beyond simple linking.^[29]

Applications

Point-in-Polygon Problem

The point-in-polygon (PIP) problem involves determining whether a given point lies inside, outside, or on the boundary of a polygonal region defined by a closed chain of line segments. One classical solution leverages the winding number of the polygon's boundary curve around the test point: if the winding number is nonzero, the point is considered inside the polygon; otherwise, it is outside.^[32] This approach naturally accounts for the topological encircling of the point by the boundary and is particularly robust for polygons with self-intersections, where it identifies regions enclosed by a net nonzero revolution of the curve.^[32] To implement this, the winding number n is computed by summing the signed angles subtended by each edge of the polygon at the test point and normalizing by $2\pi. Specifically, for a polygon with vertices p_0, p_1, \dots, p_{n-1} and test point q, the angle \phi_i for edge (p_i, p_{i+1}) (with p_n = p_0) is the oriented angle from the vector \overrightarrow{q p_i} to \overrightarrow{q p_{i+1}}, taken in (-\pi, \pi]. Then,

n = \frac{1}{2\pi} \sum_{i=0}^{n-1} \phi_i,

which yields an integer value due to the closed nature of the curve. This summation can be performed efficiently using the two-argument arctangent function to avoid branch cut issues, ensuring numerical stability even for points near edges. The method handles self-intersecting polygons by assigning nonzero winding numbers to regions where the boundary winds net positively or negatively around the point, thus delineating multiple interior components if present.^[32] For simple polygons—those without self-intersections—this test connects directly to the Jordan curve theorem, which states that a simple closed curve divides the plane into an interior and exterior region. In such cases, the absolute value of the winding number is 1 for points inside the polygon (positive for counterclockwise orientation, negative for clockwise) and 0 outside, providing a topological guarantee of the interior's boundedness.^[33] The algorithm runs in O(n) time, where n is the number of vertices, as it requires a single pass over all edges to accumulate the angles. An alternative is the even-odd rule (or ray-casting with parity), which counts boundary crossings along a ray from the test point and deems the point inside if the count is odd; however, this ignores orientation and may misclassify regions in self-intersecting or multiply connected polygons, whereas the winding number preserves directional information for more accurate handling of oriented boundaries.^[32]^[33] As a representative example, consider a counterclockwise-oriented triangle with vertices (0,0), (1,0), and (0.5, 1). For a test point inside, such as (0.5, 0.5), the signed angles from the point to the edges sum to $2\pi, yielding n = 1, confirming it is inside. For an exterior point like (2, 0.5), the angles sum to 0, so n = 0, indicating it is outside.^[33]

Proof of the Fundamental Theorem of Algebra

The topological proof of the Fundamental Theorem of Algebra utilizes the winding number to demonstrate that every non-constant polynomial with complex coefficients possesses at least one complex root. Consider a polynomial p(z) = a_k z^k + a_{k-1} z^{k-1} + \dots + a_0 of degree k \geq 1, where a_k \neq 0. Assume, for contradiction, that p(z) has no roots in the complex plane, so p(z) \neq 0 for all z \in \mathbb{C}. This assumption implies that p maps the entire complex plane into \mathbb{C} \setminus \{0\}.^[6] To proceed, parameterize circles in the complex plane as \gamma_R(t) = R e^{it} for t \in [0, 2\pi] and radius R > 0, which are closed curves. The image curve p \circ \gamma_R traces a loop in \mathbb{C} \setminus \{0\}. For sufficiently large R, specifically R > r_0 where r_0 ensures the lower-degree terms are dominated, |p(z) - a_k z^k| < |a_k z^k| on |z| = R, implying that p(z) behaves asymptotically like a_k z^k. Thus, as z traverses \gamma_R once, the argument of p(z) changes by $2\pi k, yielding a winding number n(p \circ \gamma_R, 0) = k \neq 0.^[34]^[6] In contrast, as R \to 0, the circle \gamma_R shrinks to the origin, and p \circ \gamma_R approaches the constant curve at p(0) \neq 0, which has winding number n(p \circ \gamma_0, 0) = 0. Since p has no roots, the family of loops \{ p \circ \gamma_R \mid R > 0 \} forms a free homotopy in \mathbb{C} \setminus \{0\}, and the winding number is a homotopy invariant, remaining constant across all R. This leads to a contradiction unless k = 0, meaning the polynomial must be constant. The proof implicitly relies on the maximum modulus principle to ensure p(z) does not vanish on or outside large circles, confirming the homotopy stays in \mathbb{C} \setminus \{0\}.^[34]^[6] More precisely, the winding number equality n(p \circ \gamma, 0) = \deg(p) for large circles \gamma follows from a homotopy deforming p to the leading term a_k z^k at infinity, preserving the topological degree. This approach aligns with the argument principle from complex analysis, which equates the winding number to the number of zeros inside the contour.^[34] This topological method traces back to Carl Friedrich Gauss, who provided one of the earliest rigorous topological proofs in 1816, building on his earlier but flawed attempt in 1799.^[35]

In Physics: Heisenberg Ferromagnet Equations

The Heisenberg ferromagnet model describes a continuous chain of classical spins \mathbf{S}(x,t) \in \mathbb{S}^2, where \mathbf{S} is a unit vector field representing the local magnetization direction in a one-dimensional ferromagnetic material. The dynamics are governed by the isotropic Landau-Lifshitz equation without damping,

\frac{\partial \mathbf{S}}{\partial t} = \mathbf{S} \times \frac{\partial^2 \mathbf{S}}{\partial x^2},

which arises as the continuum limit of the discrete Heisenberg Hamiltonian with nearest-neighbor exchange interactions. This equation captures nonlinear spin wave propagation and is completely integrable, as demonstrated through its equivalence to the focusing nonlinear Schrödinger equation via stereographic projection of the spin field onto the complex plane.^[36] A key conserved quantity in this model is an integer n related to the number of solitons, which characterizes the global structure of spin configurations. Using stereographic coordinates, where the spin is projected to a complex field w = (S_1 + i S_2)/(1 + S_3), the quantity n is given by

n = \frac{1}{2\pi} \int_{-\infty}^{\infty} \frac{\partial \phi}{\partial x} \, dx = \frac{\phi(+\infty) - \phi(-\infty)}{2\pi},

where \phi = \arg(w) is the azimuthal phase angle. This integer is conserved due to the integrability of the system, classifying multi-soliton solutions, though not protected by topology since \pi_1(\mathbb{S}^2) = 0. The equation of motion preserves n, ensuring stability against small perturbations within the same class. Multi-soliton configurations carry a total n equal to the sum of individual contributions, while breathers—oscillatory bound states—typically have net zero n as they involve soliton-antisoliton pairs.^[37] Soliton solutions, including magnons (linear spin waves in the small-amplitude limit) and nonlinear breathers, are classified by their n, which determine their interaction properties. Magnons correspond to single-soliton excitations with n = \pm 1, representing localized spin flips propagating along the chain. Breathers, as nonlinear superpositions, exhibit periodic oscillations in the spin profile and are labeled by pairs of n whose net value vanishes, facilitating elastic scattering without radiation. The integrability allows exact construction of these solutions using the inverse scattering transform, where the spectral parameter \lambda \in \mathbb{C} in the Lax pair encodes discrete eigenvalues corresponding to solitons, with the number and positions reflecting the total n. Geometrically, spin configurations in the model represent immersed curves on \mathbb{S}^2, with n linking the dynamics to differential geometry. This perspective highlights how time evolution preserves the invariant, analogous to conserved quantities in integrable systems. For illustration, the one-soliton solution with n=1 takes the form (in stereographic coordinates, then back-projected),

S_3(x,t) = 1 - 2 \eta^2 \sech^2 [\eta (x - v t - x_0)],

S_1 + i S_2 = 2 i \eta \sech [\eta (x - v t - x_0)] \exp[i (k x - \omega t + \phi)],

where \eta > 0 is the inverse width, v = 2 k the velocity, \omega = 2 (k^2 - \eta^2) the frequency, and parameters satisfy dispersion relations from the Lax eigenvalues; this configuration corresponds to a single winding and approaches the uniform ground state (0,0,1) at spatial infinities.^[36]

In Functional Analysis

In functional analysis, the winding number plays a central role in index theory, particularly for Fredholm operators on Hilbert spaces, where it quantifies the topological obstruction to invertibility. For an elliptic pseudodifferential operator on a compact manifold, the Fredholm index, defined as the difference between the dimensions of the kernel and cokernel, equals the winding number of the principal symbol map from the unit cotangent sphere bundle (which includes circles S¹ for one-dimensional components) to the general linear group GL(n,ℂ).^[38] This connection arises because the symbol's non-vanishing ensures Fredholmness, and the winding number captures the degree of the determinant map around the origin in ℂ*.^[39] A prominent application occurs with Toeplitz operators on the Hardy space H² of the unit disk, where the operator T_γ induced by a continuous symbol γ on the unit circle T is Fredholm if γ avoids zero, and its index is the negative of the winding number of γ: T → ℂ*.^[40] Specifically, ind(T_γ) = -wind(γ) = dim(ker T_γ) - dim(coker T_γ), reflecting how the winding measures the imbalance between holomorphic solutions in the kernel and anti-holomorphic obstructions in the cokernel.^[41] This Toeplitz index formula, established in the 1960s, extends the classical winding number from complex analysis to operator theory.^[40] The Atiyah-Singer index theorem generalizes this by equating the analytical index of elliptic operators to a topological index, which in cases reducible to circle symbols involves winding-like degrees computed via characteristic classes.^[38] For instance, the unilateral shift operator S on ℓ²(ℕ), a Toeplitz operator with symbol the identity function z on T (winding number 1), has empty kernel and one-dimensional cokernel, yielding index -1.^[42] Key properties of the winding number in this context include its integer-valued nature, as it counts net encirclements, and homotopy invariance under deformations within the space of invertible symbols, ensuring the index remains constant in connected components of the symbol algebra.^[39] These features make it a robust topological invariant for classifying Fredholm operators.^[40]

References

[1]
[PDF] MATH-UA 377 Differential Geometry Winding and Rotation Numbers ...
Feb 17, 2022 · Winding number W(p,C) of a closed planar curve C ⊂ A2 around a point p /∈ C is the number of times the curve goes counterclockwise around p.
[2]
The Winding Number - UMSL
Definition: Suppose we are given a map MATH such that MATH and such that MATH then, MATH is called the winding number or degree of $\QTR{Large}{g ...
[3]
[PDF] Math 3228 - Week 7 • Winding numbers • The argument principle
2πi. /j dz z - z 0 . The winding number is not defined if z 0 passes through 7. The winding number is always an integer, and (informally speaking) ...
[4]
02-winding-number
That is, the winding number of around any point q ∉ P is precisely the number of times that the area around is counted by the shoelace formula.
[5]
[PDF] The Fundamental Theorem of Algebra - Brown Math Department
Oct 1, 2014 · The winding number measures how many times the curve f(C) winds around the origin. This definition can be made more formal mathematically, but ...
[6]
[PDF] WINDING NUMBER AND APPLICATIONS We denote by S1 ⊂ R2 ...
We define the winding number of c by: n(c) = 1. 2π. [ϕ(2π) − ϕ(0)]. Note ... by definition c, c have the same winding number). Some applications of the ...
[7]
[PDF] 3. The shoelace formula and the winding number
The winding number is how many full turns we have done at the end, with counterclockwise turns counting as +1 and clockwise ones as −1.
[8]
[PDF] Complex Analysis Lecture Notes - UC Davis Math
Jun 15, 2021 · More generally, one can define the winding number at z = z0 as the number of times a curve γ winds around an arbitrary point z0, which (it ...<|control11|><|separator|>
[9]
A figure 8 has winding number 0, and therefore cannot bound an ...
A figure 8 has winding number 0, and therefore cannot bound an immersed disk. But a "double 8" has winding number 1, and does not bound an immersed disk either.
[10]
[PDF] Winding Numbers of Closed Path - Trinity College Dublin
your dog went around the oak tree in the centre of the park is equal to the number of times that you yourself went around that tree. Page 24. 2. Winding ...
[11]
[PDF] SMT 2021 Power Round April 17, 2021 - Stanford Math Tournament
Apr 17, 2021 · In mathematical terms, winding number is preserved under. “continuous deformation”. 2.2 An integral formula. Here, we give an explicit formula ...
[12]
Analysis - Complexity, Functions, Theory | Britannica
A proof was published by Cauchy in 1825, and this result is now named Cauchy's theorem. Cauchy went on to develop a vast theory of complex analysis and its ...
[13]
argument principle - PlanetMath.org
Mar 22, 2013 · This fact is known as the argument principle. The principle may be stated in another form which makes the origin of the name apparent: If a ...
[14]
[PDF] Cauchy's Theorem(s) - John McCuan
Mar 10, 2022 · Let us prove Goursat's theorem and Cauchy's theorem in a triangle. Proof of Goursat's theorem: Every triangular domain U can be partitioned into.
[15]
CHAPTER 7 Winding Numbers and Topology - Oxford Academic
Mar 23, 2023 · The winding number concept allows one to make the desired distinction clearly. The chapter then looks at Hopf's Degree Theorem; polynomials and ...<|control11|><|separator|>
[16]
Witold Hurewicz (1904 - 1956) - Biography - MacTutor
Witold Hurewicz made important contributions to algebraic topology including discovering higher homotopy groups. Thumbnail of Witold Hurewicz View four ...
[17]
https://mathshistory.st-andrews.ac.uk/Biographies/Hurewicz/
[18]
Winding Numbers on Discrete Surfaces - ACM Digital Library
The winding number is the number of times a curve wraps around a given point. Winding numbers are a basic component of geometric algorithms such as point-in- ...
[19]
None
Below is a merged summary of the winding number definition from Hatcher's *Algebraic Topology*, consolidating all the information from the provided segments into a single, comprehensive response. To maximize detail and clarity, I will use a table format for key information (e.g., definitions, formulas, properties) and supplement it with narrative text for context and additional details. The table will be presented in CSV-like format for density and ease of reference.
[20]
Winding Numbers and Cauchy's Theorem - SpringerLink
Lang, S. (1999). Winding Numbers and Cauchy's Theorem. In: Complex Analysis. Graduate Texts in Mathematics, vol 103. Springer, New York, NY.<|control11|><|separator|>
[21]
Influence of winding number on vortex knots dynamics - Nature
Nov 21, 2019 · The ratio w = q/p (w > 0) defines the winding number and is a measure of the knot topology. When w = q/p is irrational the curve forms a ...Missing: projection | Show results with:projection
[22]
[PDF] Differential Forms Lecture Notes Liam Mazurowski
Differential forms are a certain class of objects that can be integrated. Hence to understand differential forms it's helpful to start with the simplest ...
[23]
[PDF] 6 Differential forms
The number w(g) = 1. 2p ∫S1 g⇤w is the winding number of g. (One can show that this is always an integer, and that two loops can be deformed into each ...
[24]
[PDF] global properties of plane and space curves - UChicago Math
Rotation Index and Winding Number. Our first global theorems on plane curves come from looking at the unit tangent vector T(t) for some regular, closed curve x ...
[25]
[PDF] LECTURE-10 Index of a curve For a piecewise smooth (not ...
For a piecewise smooth (not necessarily close) curve, we defined the index or winding number around a point p /∈ γ by ... A chain is a formal linear combination ...
[26]
[PDF] differential-geometry-2024.pdf - Harvard Mathematics Department
Do Carmo, Differential Geometry of Curves and Surfaces”, 2. ... The case r(t) = [cos(nt),sin(nt)] with t ∈ [0,2π] shows that the rotation index can take any ...
[27]
[PDF] Unit 6: Hopf Umlaufsatz
To the right, a simple closed smooth curve in the plane. What is its rotation number? moves on the circle T. If we deform the curve the total change remains the ...
[28]
Mathematics of doodling and the winding number - MathOverflow
Sep 15, 2011 · The winding number is how many times a curve goes around a marked point; the turning number is how many times its velocity vector goes around the origin.
[29]
[PDF] Alexander polynomial of knots - UC Berkeley math
As noted earlier, by the Alexander duality, the first homology group of the knot complement of a tame knot is the infinite cyclic group Z, so is the ...Missing: winding | Show results with:winding<|control11|><|separator|>
[30]
[PDF] Knot complements and Spanier–Whitehead duality - Academic Web
Alexander duality implies that the homology groups of Sn \ f(X) and Sn \ g(X) are the same. Despite this isomorphism of homology groups, the spaces Sn \ f(X) ...Missing: winding number
[31]
[PDF] arXiv:2308.16853v1 [math.GT] 31 Aug 2023
Aug 31, 2023 · ... linking number, defined by considering the homology class of one in the complement of the other. First observe that the meridian of K1 ...
[32]
[PDF] Beginning Course Lecture 3
May 17, 2012 · 2.3 Linking numbers and intersection. Theorem 3. lk(K1,K2) is the signed intersection number between K2 and any Seifert sur- face for K1 ...
[33]
[PDF] The Point in Polygon Problem for Arbitrary Polygons
In Sec. 2 we explain in detail how the incremental angle algorithm [12] for determining the winding number can be derived mathematically. Further analysis of ...
[34]
[PDF] 2 Winding Numbers
If the polygon winds clockwise around o, the winding number is negative. Crucially, the winding number is only well defined if the polygon does not contain the ...
[35]
1.01 Winding numbers and the fundamental theorem of algebra
Winding numbers are topological, related to the fundamental theorem of algebra, and are homotopy invariant. The fundamental theorem states a nonconstant ...Missing: source | Show results with:source
[36]
Fund theorem of algebra - MacTutor History of Mathematics
A third proof by Gauss also in 1816 is, like the first, topological in nature. Gauss introduced in 1831 the term 'complex number'. The term 'conjugate' had been ...
[37]
[PDF] Atiyah-Singer Index theorem
The nice fact about these Fredholm operators is that the index is the winding number of g around 0. acting on complex functions is a differential operator of ...
[38]
[PDF] The Atiyah–Singer index theorem
... Fredholm for all φ ∈ C(S1,GL(V )). Show that the in- dex of Tφ coincides with the negative of the winding number of det(φ): S1 → C×. I.7.10. Exercise. Let ...
[39]
[PDF] The winding number and the Fredholm index
Aug 3, 2022 · The theorem stating this connection is called the Toeplitz index Theorem and will serve as the main theorem of this thesis.
[40]
[PDF] Winding Numbers and Toeplitz Operators
A Toeplitz Operator maps a function in to another function in by the following formula: where is the projection onto. Example: Hardy Space and Toeplitz ...
[41]
[PDF] Paris Lectures on Topological Insulators - Math (Princeton)
Its kernel is empty and the kernel of its adjoint, the left shift operator, is spanned by δ1 and is hence one dimensional. index (R) = −1 . Note that ...