John Edwin Luecke Biography

John Edwin Luecke is an American mathematician who works in topology and knot theory. He got his Ph.D. in 1985 from the University of Texas at Austin and is now a professor in the department of mathematics at that institution.

John Edwin Luecke Age

John Edwin Luecke is an American mathematician who works in topology and knot theory. He got his Ph.D. in 1985 from the University of Texas at Austin and is now a professor in the department of mathematics at that institution.

His pieces of information about the birth date, place of birth, age are not yet revealed yet but stay ready for the update soon

John Edwin Luecke Work

Luecke specializes in knot theory and 3-manifolds. In a 1987 paper Luecke, Marc Culler, Cameron Gordon, and Peter Shalen proved the cyclic surgery theorem.

In a 1989 paper, Luecke and Cameron Gordon proved that knots are determined by their complements, a result is now known as the Gordon–Luecke theorem.

Dr. Luecke received an NSF Presidential Young Investigator Award in 1992 and Sloan Foundation fellow in 1994. In 2012 he became a fellow of the American Mathematical Society.

John Edwin Luecke Wife, Married

John Edwin Luecke is an American mathematician who works in topology and knot theory. his information about Wife, marriages, wedding are not yet revealed yet but stay ready for the update soon

John Edwin Luecke Gordon–Luecke theorem

In mathematics, the Gordon–Luecke theorem on knot complements states that if the complements of two tame knots are homeomorphic, then the knots are equivalent. In particular, any homeomorphism between knot complements must take a meridian to a meridian.

The theorem is usually stated as “knots are determined by their complements”; however, this is slightly ambiguous as it considers two knots to be equivalent if there is a self-homeomorphism taking one knot to the other. Thus mirror images are neglected.

Often two knots are considered equivalent if they are isotopic. The correct version, in this case, is that if two knots have complements which are orientation-preserving homeomorphic, then they are isotopic.

These results follow from the following (also called the Gordon–Luecke theorem): no nontrivial Dehn surgery on a nontrivial knot in the 3-sphere can yield the 3-sphere.

The theorem was proved by Cameron Gordon and John Luecke. Essential ingredients of the proof are their joint work with Marc Culler and Peter Shalen on the cyclic surgery theorem, combinatorial techniques in the style of Litherland, thin position, and Scharlemann cycles.

For link complements, it is not, in fact, true that links are determined by their complements. For example, JHC Whitehead proved that there are infinitely many links whose complements are all homeomorphic to the Whitehead link.

His construction is to twist along a disc spanning an unknotted component (as is the case for either component of the Whitehead link). Another method is to twist along an annulus spanning two components.

Gordon proved that for the class of links where these two constructions are not possible there are finitely many links in this class with a given complement.

John Edwin Luecke Topology

In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumbling and bending, but not tearing or gluing. Read also about Donald Ervin Knuth

A topological space is a set endowed with a structure, called a topology, which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity.

Euclidean spaces, and, more generally, metric spaces are examples of topological spaces, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological property.

Basic examples of topological properties are the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; connectedness, which allows distinguishing a circle from two non-intersecting circles.

The ideas underlying topology go back to Gottfried Leibniz, who in the 17th century envisioned the geometric situs and analysis situs. Leonhard Euler’s Seven Bridges of Königsberg problem and polyhedron formula are arguably the field’s first theorems.

The term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed.

John Edwin Luecke Knot theory

In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined together so that it cannot be undone, the simplest knot being a ring (or “unknot”).

In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3 (in topology, a circle isn’t bound to the classical geometric concept, but to all of its homeomorphisms).

Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R3 upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.

Knots can be described in various ways. Given a method of description, however, there may be more than one description that represents the same knot. For example, a common method of describing a knot is a planar diagram called a knot diagram.

Any given knot can be drawn in many different ways using a knot diagram. Therefore, a fundamental problem in knot theory is determining when two descriptions represent the same knot.

A complete algorithmic solution to this problem exists, which has unknown complexity. In practice, knots are often distinguished by using a knot invariant, a “quantity” which is the same when computed from different descriptions of a knot. Important invariants include knot polynomials, knot groups, and hyperbolic invariants.

The original motivation for the founders of knot theory was to create a table of knots and links, which are knots of several components entangled with each other. More than six billion knots and links have been tabulated since the beginnings of knot theory in the 19th century.

To gain further insight, mathematicians have generalized the knot concept in several ways. Knots can be considered in other three-dimensional spaces and objects other than circles can be used; see knot (mathematics). Higher-dimensional knots are n-dimensional spheres in m-dimensional Euclidean space.