with p strands. The appropriate braid word is (This formula assumes the common convention that braid generators are right twists, which is not followed by the Wikipedia page on braids.) The crossing number of a (p,q) torus knot with p,q > 0 is given by c = min((p−1)q, (q−1)p). The genus of a torus knot with p,q > 0 is The Alexander polynomial of a torus knot is where The Jones polynomial of a (right-handed) torus knot is given by The complement of a torus knot in the 3-sphere is a Seifert-fibered manifold, fibred over the disc with two singular fibres. Let Y be the p-fold dunce cap with a disk removed from the interior, Z be the q-fold dunce cap with a disk removed from its interior, and X be the quotient space obtained by identifying Y and Z along their boundary circle. The knot complement of the (p, q) -torus knot deformation retracts to the space X. Therefore, the knot group of a torus knot has the presentation Torus knots are the only knots whose knot groups have nontrivial center (which is infinite cyclic, generated by the element in the presentation above). The stretch factor of the (p,q) torus knot, as a curve in Euclidean space, is Ω(min(p,q)), so torus knots have unbounded stretch factors. Undergraduate researcher John Pardon won the 2012 Morgan Prize for his research proving this result, which solved a problem originally posed by Mikhail Gromov. Connection to complex hypersurfaces The (p,q)−torus knots arise when considering the link of an isolated complex hypersurface singularity. One intersects the complex hypersurface with a hypersphere, centred at the isolated singular point, and with sufficiently small radius so that it does not enclose, nor encounter, any other singular points. The intersection gives a submanifold of the hypersphere. Let p and q be coprime integers, greater than or equal to two. Consider the holomorphic function given by Let be the set of such that Given a real number we define the real three-sphere as given by The function has an isolated critical point at since if and only if Thus, we consider the structure of close to In order to do this, we consider the intersection This intersection is the so-called link of the singularity The link of , where p and q are coprime, and both greater than or equal to two, is exactly the (p,q)−torus knot. List The