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Solid torus

A solid torus is a three-dimensional manifold with boundary, geometrically formed as a solid of revolution by rotating a filled disk of radius r (the minor radius) around an external axis at a distance R (the major radius) from its center, producing a doughnut-like shape where R > r > 0.^[1]^[2] Topologically, it is defined as a space homeomorphic to the Cartesian product S^1 \times D^2, where S^1 is the circle and D^2 is the closed two-dimensional disk, making its boundary an ordinary torus surface homeomorphic to S^1 \times S^1.^[3] The volume of this solid is given by the formula V = 2\pi^2 R r^2, which can be derived using the method of disks or Pappus's centroid theorem.^[1]^[2] In differential geometry and calculus, the solid torus serves as a model for computing volumes of revolution and studying parametric surfaces, with coordinates often parameterized as (x,y,z) = ((R + r \cos \theta) \cos \phi, (R + r \cos \theta) \sin \phi, r \sin \theta) for $0 \leq \theta, \phi \leq 2\pi.^[1] Its surface area, focusing on the boundary torus, is $4\pi^2 R r, highlighting its role in integration techniques like the shell or washer methods.^[1] The solid torus holds central importance in three-manifold topology, where it appears as a building block for more complex spaces; for instance, every embedded torus in the three-sphere S^3 bounds a solid torus on at least one side, a fact underpinning the classification of knots and links via Dehn surgery, which replaces the interior of a solid torus neighborhood of a knot with another solid torus glued along a specified curve.^[4] It is also fundamental in Seifert-fibered spaces, constructed by attaching solid tori to circle bundles over surfaces with specified exceptional fibers, aiding the decomposition of irreducible three-manifolds into atoroidal or Seifert components.^[4] These properties make the solid torus indispensable for understanding incompressible surfaces and the geometrization of three-manifolds.^[4]

Definition and Construction

Formal Definition

The solid torus, denoted T, is formally defined in topology as the Cartesian product of a closed 2-dimensional disk D^2 and a circle S^1, yielding T = D^2 \times S^1.^[5] This construction equips T with the product topology, making it a compact 3-dimensional space.^[4] The boundary of the solid torus is the torus surface \partial T = S^1 \times S^1, which distinguishes it from the hollow torus consisting solely of this boundary surface.^[6] As a compact 3-manifold with boundary, the solid torus serves as a fundamental building block in the study of 3-dimensional manifolds, particularly in decompositions and surgeries.^[4] In its standard embedding in Euclidean 3-space \mathbb{R}^3, the solid torus is realized without self-intersection using toroidal coordinates (r, \theta, \phi), where r ranges from 0 to the minor radius a (the radius of the tubular cross-section), $0 \leq \theta < 2\pi is the poloidal angle, and $0 \leq \phi < 2\pi is the toroidal angle, with the major radius R (distance from the center of the tube to the center of the torus) satisfying a < R to ensure the embedding is disjoint from itself.^[1]

Geometric Constructions

One common geometric construction of the solid torus in Euclidean 3-space \mathbb{R}^3 involves generating a solid of revolution by rotating a filled disk around an external axis. Specifically, consider a disk of radius a centered at a distance R > a from the axis of rotation; revolving this disk around the axis produces a solid torus with major radius R and minor radius a.^[7] This method embeds the solid torus standardly in \mathbb{R}^3, where the core circle lies along the path traced by the disk's center. Another approach realizes the solid torus as a tubular neighborhood of an embedded circle in \mathbb{R}^3. For any smoothly embedded circle K \subset \mathbb{R}^3, a sufficiently small \epsilon-neighborhood N_\epsilon(K) forms an open solid torus with K as its core curve, provided \epsilon is chosen small enough to avoid self-intersections.^[8] This construction generalizes to knotted cores, yielding non-standard embeddings while preserving the topology of the solid torus.^[9] The solid torus can also be constructed as a quotient space by starting with a cylinder [0,1] \times S^1 and capping its boundary components with disks. Gluing a 2-disk to each end of the cylinder via the identity map on the boundary circles S^1 \times \{0\} and S^1 \times \{1\} yields S^1 \times D^2, the solid torus.^[9] More generally, quotient constructions arise in 3-manifold decompositions, such as forming lens spaces by identifying boundaries of two solid tori via a specific diffeomorphism.^[4] A construction involves the Clifford torus in the 3-sphere S^3. The Clifford torus, embedded as the flat submanifold \{(z_1, z_2) \in \mathbb{C}^2 : |z_1| = |z_2| = 1/\sqrt{2}\} in S^3 \subset \mathbb{C}^2, divides S^3 into two congruent solid tori.^[10] Stereographic projection from S^3 to \mathbb{R}^3 maps these solid tori to regions in Euclidean space, providing the standard embedded solid torus.^[1]

Geometric Properties

Parametric Representation

The solid torus in \mathbb{R}^3 admits a standard parametric representation using the minor radial distance r, the poloidal angle \theta, and the toroidal angle \phi. The coordinates are mapped to Cartesian coordinates via

\begin{align*} x &= (R + r \cos \theta) \cos \phi, \\ y &= (R + r \cos \theta) \sin \phi, \\ z &= r \sin \theta, \end{align*}

where R > 0 is the major radius, a > 0 is the minor radius with r \in [0, a], and \theta, \phi \in [0, 2\pi).^[1] This parameterization describes the solid torus as the set of all points obtained by revolving a filled disk of radius a centered at (R, 0, 0) around the z-axis. The Jacobian determinant of this transformation, \left| \det \frac{\partial (x, y, z)}{\partial (r, \theta, \phi)} \right| = r (R + r \cos \theta), provides the volume element dV = r (R + r \cos \theta) \, dr \, d\theta \, d\phi for integration over the solid torus.^[1] Integrating this yields the volume V = 2\pi^2 R a^2.^[1] The inverse transformation from Cartesian coordinates (x, y, z) to the parametric coordinates is obtained as follows: \phi = \atantwo(y, x), \rho = \sqrt{x^2 + y^2}, r = \sqrt{(\rho - R)^2 + z^2}, and \theta = \atantwo(z, \rho - R), where \atantwo accounts for the correct quadrant.^[1] An alternative parametric representation uses orthogonal toroidal coordinates (\sigma, \tau, \phi), which are suited for regions bounded by toroidal surfaces. These are defined by

\begin{align*} x &= \frac{\alpha \sinh \tau \cos \phi}{\cosh \tau - \cos \sigma}, \\ y &= \frac{\alpha \sinh \tau \sin \phi}{\cosh \tau - \cos \sigma}, \\ z &= \frac{\alpha \sin \sigma}{\cosh \tau - \cos \sigma}, \end{align*}

where \alpha > 0 is the focal ring radius (scale parameter), \sigma \in [0, 2\pi), \tau \in [\tau_0, \infty), and \phi \in [0, 2\pi), with \tau_0 > 0 chosen such that the boundary surface \tau = \tau_0 is the torus of major radius R = \alpha \coth \tau_0 and minor radius a = \alpha / \sinh \tau_0.^[11] In this system, the cylindrical radius \rho = \sqrt{x^2 + y^2} ranges over approximately [R - a, R + a] within the solid torus. The scale factors for the metric in toroidal coordinates are h_\sigma = h_\tau = \alpha / (\cosh \tau - \cos \sigma) and h_\phi = \alpha \sinh \tau / (\cosh \tau - \cos \sigma).^[11] The Jacobian determinant, which is the product of the scale factors, is \alpha^3 \sinh \tau / (\cosh \tau - \cos \sigma)^3, yielding the volume element dV = [\alpha^3 \sinh \tau / (\cosh \tau - \cos \sigma)^3] \, d\sigma \, d\tau \, d\phi for volume integration.^[11] The inverse transformation in toroidal coordinates involves solving the system for \sigma, \tau, \phi from (x, y, z), typically via \phi = \atantwo(y, x) followed by algebraic manipulation using the identities \rho = \sqrt{x^2 + y^2} and relations like \sigma = \arccos[(\rho \cosh \tau - \alpha)/(\rho - \alpha \cosh \tau / \rho)] and solving a quadratic for \tau, though explicit closed forms are complex and often computed numerically.^[11]

Metrics and Curvature

The solid torus, when embedded in Euclidean 3-space \mathbb{R}^3, inherits the induced metric from the ambient Euclidean metric via its standard parametrization. In toroidal coordinates (r, \theta, \phi), where $0 \leq r \leq a is the radial distance from the central circle of radius R, \theta \in [0, 2\pi) parametrizes the tube cross-section, and \phi \in [0, 2\pi) winds around the major axis, the line element is

ds^2 = dr^2 + r^2 d\theta^2 + (R + r \cos \theta)^2 d\phi^2.

This metric reflects the geometry of the embedding, with the d\phi^2 term varying due to the offset from the axis of rotation.^[12] The volume of the solid torus is obtained by integrating the volume form derived from this metric over the parameter domain, yielding V = 2\pi^2 R a^2. This formula arises from the product of the circumference of the major circle $2\pi R and the area of the disk cross-section \pi a^2, adjusted by the rotational sweep, and can be verified using Pappus's centroid theorem or direct integration in toroidal coordinates.^[1] The boundary of the solid torus is a toroidal surface, whose area is computed by restricting the metric to r = a and integrating the induced area element, resulting in A = 4\pi^2 R a. This represents the total surface area enclosing the solid, combining the contributions from the outer and inner equatorial bands.^[1] On this boundary surface, the Gaussian curvature K measures the intrinsic bending and is given by

K = \frac{\cos \theta}{a (R + a \cos \theta)},

which changes sign: positive on the outer equator (where \cos \theta > 0), negative on the inner equator (where \cos \theta < 0), and zero along the top and bottom circles. The mean curvature H, an extrinsic measure averaging the principal curvatures, is

H = -\frac{R + 2a \cos \theta}{2a (R + a \cos \theta)},

which vanishes only at specific points where the surface is balanced between convex and concave bending, such as along certain meridians depending on the aspect ratio R/a > 1. These curvatures highlight the non-uniform geometry of the embedded torus, distinguishing it from flat metrics.^[1] In higher dimensions, the flat torus metric on T^n = (S^1)^n is the product of standard circle metrics, ds^2 = \sum_{i=1}^n (r_i d\phi_i)^2 with constant radii r_i, yielding zero sectional curvature everywhere and enabling isometric embeddings into \mathbb{R}^{2n} but not smoothly into lower dimensions like \mathbb{R}^3 for n=2. In contrast, the curved metric induced by embedding a higher-dimensional analogue (e.g., a Clifford torus in S^3 \subset \mathbb{R}^4) preserves the flat intrinsic geometry with constant Gaussian curvature K = 0 in the induced round metric, illustrating how embedding choices can realize the flat case smoothly in higher dimensions.^[13]^[14]

Topological Properties

Manifold Structure

The solid torus T, defined as the product D^2 \times S^1, is a compact orientable 3-manifold with boundary.^[15] Its boundary \partial T is homeomorphic to the 2-torus T^2 = S^1 \times S^1, obtained as the product of the boundary circle \partial D^2 = S^1 with S^1.^[4] This structure classifies T as an orientable manifold of dimension 3, where orientability follows from the product of orientable components D^2 and S^1.^[15] The smooth structure on T arises from the product smooth structures on D^2 and S^1, equipped with their standard atlases of smooth charts.^[15] Specifically, an atlas for T consists of charts derived from local coordinates on D^2 (polar or Cartesian) and angular coordinates on S^1, ensuring transition maps are smooth diffeomorphisms.^[15] This endows T with a C^\infty differentiable structure compatible with its topological manifold properties. As a manifold, T satisfies the local Euclidean property: every interior point admits a neighborhood homeomorphic to \mathbb{R}^3, while boundary points have neighborhoods homeomorphic to the closed half-space \mathbb{H}^3 = \{ (x,y,z) \in \mathbb{R}^3 \mid z \geq 0 \}.^[15] Interior charts use open sets of the form U \times V, where U \subset \operatorname{int} D^2 is open in \mathbb{R}^2 and V \subset S^1 is an open arc homeomorphic to (0,1), mapped diffeomorphically to \mathbb{R}^3. Near the boundary, charts restrict to sets like (\partial D^2 \times [0,\epsilon)) \times V, projecting to half-space coordinates via boundary-fitted maps that preserve smoothness.^[15] The solid torus admits a handlebody decomposition as a genus-1 handlebody, constructed by attaching a single 1-handle to a 0-handle (a 3-ball B^3).^[16] This involves selecting two disjoint disks on \partial B^3, removing their interiors, and attaching a product [0,1] \times D^2 along the resulting boundary circles via an orientation-reversing map, yielding T up to homeomorphism.^[16] This decomposition highlights T as the simplest non-trivial bounded 3-manifold in handlebody theory.^[16]

Homotopy and Homology Groups

The solid torus T = S^1 \times D^2, where D^2 is the closed 2-dimensional disk, has fundamental group \pi_1(T) \cong [\mathbb{Z}](/page/Z), generated by the core circle, which corresponds to the longitude class on the boundary torus \partial T = S^1 \times S^1.^[4] The meridian class, which bounds a disk in T, is trivial in \pi_1(T).^[4] Since T deformation retracts onto its core circle S^1, it is homotopy equivalent to S^1, implying that the higher homotopy groups vanish: \pi_n(T) = 0 for all n \geq 2.^[15] This makes T an aspherical space, with all non-trivial homotopy concentrated in dimension 1.^[15] The homology groups of T can be computed using the Künneth theorem for the product structure S^1 \times D^2, where H_*(D^2; \mathbb{Z}) = \mathbb{Z} in degree 0 and 0 otherwise. This yields H_0(T; \mathbb{Z}) \cong \mathbb{Z}, H_1(T; \mathbb{Z}) \cong \mathbb{Z} (generated by the longitude), and H_n(T; \mathbb{Z}) = 0 for n \geq 2.^[15] Alternatively, the Mayer-Vietoris sequence applied to a decomposition of T into overlapping solid cylinders confirms these groups, with the sequence reducing to isomorphisms reflecting the single generator in degree 1.^[15] The relative homology H_*(T, \partial T; \mathbb{Z}) captures the interior structure relative to the boundary: H_1(T, \partial T; \mathbb{Z}) = 0, H_2(T, \partial T; \mathbb{Z}) \cong \mathbb{Z} (generated by the class of a meridional disk), and H_3(T, \partial T; \mathbb{Z}) \cong \mathbb{Z} (the relative fundamental class).^[4] From the long exact sequence of the pair (T, \partial T), the connecting homomorphism \partial: H_2(T, \partial T) \to H_1(\partial T) sends the meridional disk class to the meridian class, which is trivial in the absolute homology H_1(T).^[15] The Euler characteristic of T is \chi(T) = 0, computed as the alternating sum of Betti numbers: \chi(T) = \dim H_0(T) - \dim H_1(T) + \dim H_2(T) - \dim H_3(T) = 1 - 1 + 0 - 0.^[15] This value aligns with T being a 3-manifold homotopy equivalent to a circle.^[4]

Embeddings and Applications

Spatial Embeddings

The solid torus admits a standard unknotted embedding in 3-dimensional Euclidean space \mathbb{R}^3 as the region consisting of all points at a distance at most a from a circle of radius R lying in the xy-plane and centered at the origin, where R > a ensures the embedding is without self-intersections. This construction, often visualized as a doughnut shape, positions the core circle—the image of \{0\} \times S^1—as an unknotted loop, with the disk factor D^2 filling the tubular neighborhood around it. Such embeddings are smooth and can be parameterized using toroidal coordinates, referencing the parametric form of the boundary torus for the surface case.^[1] More generally, embeddings of the solid torus in \mathbb{R}^3 can be knotted, where the image of the core circle \{0\} \times S^1 realizes a non-trivial knot type.^[17] For instance, a trefoil knotted solid torus arises by taking a small tubular neighborhood around a trefoil knot embedded in \mathbb{R}^3, preserving the solid torus topology while the core becomes knotted.^[18] These knotted embeddings maintain the manifold structure of D^2 \times S^1 but introduce complexity in their spatial placement, with the boundary torus inheriting the knotting from the core. The complement \mathbb{R}^3 minus an unknotted solid torus embedding is homeomorphic to another solid torus, reflecting the duality seen in the 3-sphere where S^3 decomposes as the union of two solid tori along their boundary tori.^[4] This homeomorphism holds up to compactification, as the unbounded nature of \mathbb{R}^3 corresponds to placing the "missing point" of S^3 in the complement, yielding an open solid torus structure.^[4] Isotopy classes of solid torus embeddings in \mathbb{R}^3 are classified primarily by the knot type of the core curve, with additional distinction by the framing (the homotopy class of the longitude relative to the knot complement).^[18] For unknotted cores, embeddings are unique up to isotopy, while knotted cores yield distinct classes corresponding to satellite constructions around the core knot.^[19] Certain isotopies of these embeddings, particularly those involving the boundary torus, can produce ribbon knots when the boundary is pushed inward in the 4-ball B^4, creating a ribbon disk bounded by the core with only saddle singularities.^[20] In the context of spatial graph embeddings, fat vertices—thickened representations of graph vertices to avoid singularities—are realized as embedded solid tori in \mathbb{R}^3.^[21] This construction allows graphs to be smoothly embedded by replacing point vertices with small solid tori, whose cores align with incident edges, ensuring the overall embedding remains tame and without self-intersections.^[22] Such realizations are essential in 3-manifold decompositions, where the solid tori model vertex neighborhoods in intersection graphs of surfaces.^[23]

Role in Knot Theory and Dehn Surgery

In knot theory, a fundamental role of the solid torus arises as a regular neighborhood of any knot embedded in the 3-sphere S^3. For any knot K \subset S^3, there exists a tubular neighborhood N(K) homeomorphic to a solid torus S^1 \times D^2, where the core S^1 \times \{0\} is isotopic to K.^[24] This neighborhood is obtained by taking points sufficiently close to K in the standard metric on S^3, ensuring the boundary \partial N(K) is a torus framing K.^[25] The complement of a knot K in S^3, denoted S^3 \setminus \operatorname{int}(N(K)), is an open 3-manifold with boundary a torus. For the unknot, this complement is homeomorphic to a solid torus, as S^3 decomposes as the union of two solid tori glued along their boundaries. In general, for nontrivial knots, the complement is not a solid torus but a more complex manifold, often hyperbolic, whose topology encodes the knot's invariants.^[25] Dehn surgery on a knot K \subset S^3 involves removing the interior of its solid torus neighborhood N(K) to obtain the knot complement M = S^3 \setminus \operatorname{int}(N(K)), then attaching a new solid torus S^1 \times D^2 via a homeomorphism of the boundary tori that maps a simple closed curve of slope p/q \in \mathbb{Q} \cup \{\infty\} to the meridian of the new solid torus.^[26] Here, the slope is measured with respect to the standard meridian \mu and longitude \lambda on \partial M, and \infty-surgery corresponds to the original S^3. This construction yields closed 3-manifolds; for example, p/q-surgery on the unknot produces the lens space L(p,q).^[27] More generally, surgeries on nontrivial knots can produce lens spaces, Seifert fibered spaces, or hyperbolic manifolds, depending on the knot and slope.^[28] The fundamental group of the resulting manifold after p/q-Dehn surgery is the quotient of the knot group \pi_1(M) by the normal closure of the element \mu^p \lambda^q = 1, where \mu generates the infinite cyclic peripheral subgroup corresponding to the meridian.^[29] This relation reflects the disk bounded by the image of \mu^p \lambda^q in the attached solid torus, whose fundamental group is \mathbb{Z} generated by its longitude.^[25] In Thurston's geometrization conjecture, resolved by Perelman, the solid torus plays a key role through Dehn filling on cusped hyperbolic 3-manifolds, such as knot complements. For a hyperbolic knot complement with toroidal cusp, all but finitely many Dehn fillings yield hyperbolic structures on the resulting closed manifold, parametrizing a neighborhood of complete hyperbolic metrics near the cusp.^[6] This hyperbolic Dehn filling theorem ensures that most surgeries preserve hyperbolicity, linking knot theory to the geometric decomposition of 3-manifolds.^[6] Historically, Dale Rolfsen's work in the 1960s advanced the understanding of Dehn surgery by developing tables of knots and their surgeries, facilitating the classification of resulting 3-manifolds and highlighting the solid torus's utility in computational topology.^[30]

References

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### Definition and Key Properties of the Solid Torus
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One can, of course, construct these manifolds using surgeries permitted in the Kirby calculus, but one is obliged to use different knots. Kn for each n, which ...<|separator|>