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In an irreducible three-manifold with incompressible boundary, a minimizing surface can be cut along a separating incompressible torus after removing inessential intersection circles. The essential pieces have no disk or sphere components, so their ${\chi }_{-}$ costs add. Minimizing independently in the pieces and gluing their matching boundary curves gives the corresponding upper bound. This allows Thurston norm calculations for satellite knots.

A toroid is a ring-shaped surface of revolution obtained by rotating a closed planar profile about an axis in its plane that stays outside the profile. A circular profile gives a torus; a rectangular profile gives a ring with rectangular cross-section. The enclosed solid supplies the core geometry in a toroidal winding, while the mathematical surface is its boundary. The distinction matters when computing a surface area versus a volume.

In 1961, Raymond Rappaport gently removed membrane and jelly from sand dollar ova with a constricted pipette. After fertilization, he used an inverted optical microscope and rack-and-pinion manipulator to press a needle-mounted glass sphere through an egg onto the chamber floor, producing a torus-shaped biological cell. Visible spindle apparatus positions and cleavage furrow progression were the readout.^[ref]

First cytokinesis opened one side into a horseshoe containing two cell nuclei. During the next division, daughters separated from its free ends; an additional cleavage furrow crossed between mitotic asters of different spindle apparatuses. Control eggs supplied the timing comparison. This region lacked intervening chromosomes or a previous nearby spindle apparatus. This supported mitotic aster-associated positioning, without identifying the signal or establishing universal necessity.^[ref]

For nonzero coprime integers $p,q$, the torus knot $T_{p,q}$ lies on a standard unknotted torus in the three-sphere and winds $p$ times meridionally and $q$ times longitudinally. Here a meridian of a solid torus goes around the small cross-section of the doughnut, while the longitudinal direction goes around its central hole. The integers count signed turns, so changing one sign gives the mirror of a link of the torus knot. In this article $p$ is always the meridional count and $q$ the longitudinal count.

An explicit parametric curve in three-dimensional Euclidean space iswhere $R>r>0$. Its cylindrical radius $\rho =\sqrt{x^2+y^2}$ satisfies $(\rho -R{)}^2+z^2=r^2$, the equation of the supporting torus. Its meridional angle is $pt$ and its longitudinal angle is $qt$: this checks the convention directly rather than relying on the picture. Its squared speed isIf two parameter values give the same point, their difference $\delta$ satisfies $p\delta ,q\delta \in 2\pi \mathbb{Z}$. Bézout's identity for the coprime integers $p,q$ then gives $\delta \in 2\pi \mathbb{Z}$. Thus the loop closes and is an injective function on the parameter circle, with nonzero derivative; it is a smooth embedding.

For example, $R=2$, $r=0.7$, $p=2$, $q=3$ produces a trefoil knot. During one circuit the meridional angle changes by $4\pi$ and the longitudinal angle by $6\pi$. The supporting torus is a guide for construction, not a constraint on later ambient isotopies: the knot may move off it. This family makes arithmetic visible as geometry: the coprime integers determine whether a single traversal visits the whole loop without repetition.

The coprime integers hypothesis matters. If the greatest common divisor is $d=\gcd (|p|,|q|)>1$, the same parametric curve retraces a single component $d$ times; it does not automatically draw $d$ different components. The corresponding torus link is instead the full set of points on the torus whose angles satisfy $q\varphi -p\theta \in 2\pi \mathbb{Z}$, with meridional angle $\varphi$ and longitudinal angle $\theta$. It has $d$ disjoint components, each with winding counts $(p/d,q/d)$. For instance, the torus link $T_{2,2}$ consists of two parallel $(1,1)$ loops and is a Hopf link, whereas substituting $(2,2)$ into ${\gamma }_{p,q}$ simply traces one of those loops twice.

The mapping torus of a torus map with this monodromy has fundamental group generated by $x,y,t$ with $x$ central and $ty{t}^{-1}=x^{b}y$. Normal forms $x^m{y}^n{t}^k$ multiply byThis proves torsion-freeness and nilpotence of class at most two. For $b\ne 0$ its center is $⟨x⟩$ and commutator subgroup $⟨x^b⟩$; for $b=0$ it is ${\mathbb{Z}}^3$. These realize all noncyclic torsion-free nilpotent closed orientable three-manifold groups.

A vertical surface is a union of Seifert fibers over an arc or a closed curve in the base orbifold. Typical connected examples are an annulus and a torus.

For a representation on which a commuting torus or Cartan action is semisimple, the weight decomposition is the direct sum of simultaneous weight spaces. A finite-dimensional representation of a compact torus can be made unitary by Haar measure averaging; its commuting unitary matrices are simultaneously diagonalizable. Each diagonal character is an integral weight of a representation. The decomposition makes tensor and exterior-power weights additive. For nonsemisimple actions, simultaneous generalized eigenspaces give a generalized weight decomposition instead.

In the symmetric power ${\mathrm{Sym}}^m{\mathbb{C}}^n$ of the defining unitary group representation, diagonal torus weights are indexed by nonnegative integer tuples summing to $m$, with one monomial per weight space. Torus averaging extracts a monomial from any nonzero invariant complex subspace. The complexified Lie algebra acts by $E_{ij}=x_i{\partial }_{x_j}$ on the symmetric algebra. Whenever $a_j>0$, the displayed action moves one unit of degree from $j$ to $i$ with nonzero coefficient. Finite transfers connect every degree-$m$ monomial, so the subspace is the entire representation. The case $m=0$ is the one-dimensional trivial representation. This proof needs neither the Weyl character formula nor the Weyl dimension formula.

For a compact Lie group that may be disconnected, fix a maximal torus $T$ and define the normalizer quotient $N_G(T)/T$. It is finite. To prove this, identify $T$ with its Lie algebra modulo the kernel lattice of its exponential map. A torus automorphism preserves this lattice. Its differential is therefore an integral invertible matrix in a lattice basis. A connected subgroup of the normalizer induces a continuous family of such matrices, hence acts trivially on $T$.

The identity component $N_G(T{)}^0$ thus centralizes $T$ and lies in $G^0$. The maximal-torus centralizer theorem for the connected compact group gives $N_G(T{)}^0=T$. The quotient is consequently discrete and compact, hence finite. Its action on $T$ need not be faithful when $G$ is disconnected: for a finite group, $T$ is trivial and $N_G(T)/T=G$. This is why the quotient $N_G(T)/C_G(T)$ must not silently replace $N_G(T)/T$ in this convention.

For the unitary group, the diagonal maximal torus has coordinates $z_i=e^{i{\theta }_i}$ and Weyl group $S_n$. The Weyl integration formula weights independent uniform angles by $n{!}^{-1}{\prod }_{i<j}|z_i-z_j{|}^2$. The Vandermonde determinant $\det (z_i^{n-j})$ has squared torus integral $n!$, by Fourier orthogonality. Multiplying an irreducible character of highest weight $\lambda$ by this determinant gives an alternating Laurent polynomial. Its leading alternant has coefficient one; character orthogonality for compact groups forces every other alternant coefficient to vanish. This yields the quotient $\det (z_i^{{\lambda }_j+n-j})/\det (z_i^{n-j})$.