John Edwin Luecke | |
---|---|

Nationality | American |

Alma mater | University of Texas at Austin |

Known for | Gordon–Luecke theorem |

Scientific career | |

Fields | topology, knot theory |

Doctoral advisor | Cameron McAllan Gordon |

**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.

Luecke specializes in knot theory and 3-manifolds. In a 1987 paper^{ [1] } Luecke, Marc Culler, Cameron Gordon, and Peter Shalen proved the cyclic surgery theorem. In a 1989 paper^{ [2] } Luecke and Cameron Gordon proved that knots are determined by their complements, a result now known as the Gordon–Luecke theorem.

Dr Luecke received a NSF Presidential Young Investigator Award ^{ [3] }^{ [4] } in 1992 and Sloan Foundation fellow^{ [5] } in 1994. In 2012 he became a fellow of the American Mathematical Society.^{ [6] }

**William Paul Thurston** was an American mathematician. He was a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds. From 2003 until his death he was a professor of mathematics and computer science at Cornell University.

In mathematics, a **knot** is an embedding of a topological circle *S*^{1} in 3-dimensional Euclidean space, **R**^{3}, considered up to continuous deformations (isotopies).

In mathematics, the **knot complement** of a tame knot *K* is the space where the knot isn't. If a knot is embedded in the 3-sphere, then the complement is the 3-sphere minus the space near the knot. To make this precise, suppose that *K* is a knot in a three-manifold *M*. Let *N* be a tubular neighborhood of *K*; so *N* is a solid torus. The knot complement is then the complement of *N*,

**Peter B. Shalen** is an American mathematician, working primarily in low-dimensional topology. He is the "S" in JSJ decomposition.

In mathematics, the **Smith conjecture** states that if *f* is a diffeomorphism of the 3-sphere of finite order, then the fixed point set of *f* cannot be a nontrivial knot.

**Cameron Gordon** is a Professor and Sid W. Richardson Foundation Regents Chair in the Department of Mathematics at the University of Texas at Austin, known for his work in knot theory. Among his notable results is his work with Marc Culler, John Luecke, and Peter Shalen on the cyclic surgery theorem. This was an important ingredient in his work with Luecke showing that knots were determined by their complement. Gordon was also involved in the resolution of the Smith conjecture.

**John Willard Morgan** is an American mathematician known for his contributions to topology and geometry. He is a Professor Emeritus at Columbia University and a member of the Simons Center for Geometry and Physics at Stony Brook University.

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.

**William Bernard Raymond Lickorish** is a mathematician. He is emeritus professor of geometric topology in the Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, and also an emeritus fellow of Pembroke College, Cambridge. His research interests include topology and knot theory. He was one of the discoverers of the HOMFLY polynomial invariant of links, and proved the Lickorish-Wallace theorem which states that all closed orientable 3-manifolds can be obtained by Dehn surgery on a link.

In the mathematical theory of knots, a **Berge knot** or **doubly primitive knot** is any member of a particular family of knots in the 3-sphere. A Berge knot *K* is defined by the conditions:

*K*lies on a genus two Heegaard surface*S*- in each handlebody bound by
*S*,*K*meets some meridian disc exactly once.

Knots have been used for basic purposes such as recording information, fastening and tying objects together, for thousands of years. The early, significant stimulus in knot theory would arrive later with Sir William Thomson and his vortex theory of the atom.

**Mladen Bestvina** is a Croatian-American mathematician working in the area of geometric group theory. He is a Distinguished Professor in the Department of Mathematics at the University of Utah.

**Marc Edward Culler** is an American mathematician who works in geometric group theory and low-dimensional topology. A native Californian, Culler did his undergraduate work at the University of California at Santa Barbara and his graduate work at Berkeley where he graduated in 1978. He is now at the University of Illinois at Chicago. Culler is the son of Glen Jacob Culler who was an important early innovator in the development of the Internet.

In three-dimensional topology, a branch of mathematics, the **cyclic surgery theorem** states that, for a compact, connected, orientable, irreducible three-manifold *M* whose boundary is a torus *T*, if *M* is not a Seifert-fibered space and *r,s* are slopes on *T* such that their Dehn fillings have cyclic fundamental group, then the distance between *r* and *s* is at most 1. Consequently, there are at most three Dehn fillings of *M* with cyclic fundamental group. The theorem appeared in a 1987 paper written by Marc Culler, Cameron Gordon, John Luecke and Peter Shalen.

The **Presidential Young Investigator Award** (PYI) was awarded by the National Science Foundation of the United States Federal Government. The program operated from 1984 to 1991, and was replaced by the NSF Young Investigator (NYI) Awards and Presidential Faculty Fellows Program (PFF).

In mathematical knot theory, a **Conway sphere**, named after John Horton Conway, is a 2-sphere intersecting a given knot or link in a 3-manifold transversely in four points. In a knot diagram, a Conway sphere can be represented by a simple closed curve crossing four points of the knot, the cross-section of the sphere; such a curve does not always exist for an arbitrary knot diagram of a knot with a Conway sphere, but it is always possible to choose a diagram for the knot in which the sphere can be depicted in this way. A Conway sphere is *essential* if it is incompressible in the knot complement. Sometimes, this condition is included in the definition of Conway spheres.

**Jonathan Micah Rosenberg** is an American mathematician, working in algebraic topology, operator algebras, K-theory and representation theory, with applications to string theory in physics.

**Thomas** "**Tim**" **Daniel Cochran** was a professor of Mathematics at Rice University specializing in topology, especially low-dimensional topology, the theory of knots and links and associated algebra.

**Lisa Marie Piccirillo** is an American mathematician who works on geometry and low-dimensional topology. In 2020, Piccirillo published a mathematical proof in the journal *Annals of Mathematics* determining that the Conway knot is not a slice knot, answering an unsolved problem in knot theory first proposed over fifty years prior by English mathematician John Horton Conway. In July 2020, she became an assistant professor of mathematics at Massachusetts Institute of Technology.

- ↑ M. Culler, C. Gordon, J. Luecke, P. Shalen (1987). Dehn surgery on knots. The Annals of Mathematics (
*Annals of Mathematics*) 125**(2)**: 237-300. - ↑ Cameron Gordon and John Luecke,
*Knots are determined by their complements*. J. Amer. Math. Soc. 2 (1989), no. 2, 371–415. - ↑ The University of Texas at Austin; Faculty profile Archived 2011-09-27 at the Wayback Machine
- ↑ NSF Presidential and Honorary Awards
- ↑ Sloan Research Fellowships
^{[ permanent dead link ]} - ↑ List of Fellows of the American Mathematical Society, retrieved 2013-02-02.

- John Edwin Luecke at the Mathematics Genealogy Project
- Luecke's home page at the University of Texas at Austin

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