Ralph Fox
Ralph Fox  

Born 
Morrisville, Pennsylvania 
March 24, 1913
Died  December 23, 1973 Philadelphia 
(aged 60)
Nationality  American 
Alma mater  
Known for 

Scientific career  
Fields  Mathematics 
Doctoral advisor  Solomon Lefschetz 
Doctoral students 
Ralph Hartzler Fox (March 24, 1913 – December 23, 1973) was an American mathematician. As a professor at Princeton University, he taught and advised many of the contributors to the Golden Age of differential topology, and he played an important role in the modernization and mainstreaming of knot theory.
Biography
Ralph Fox attended Swarthmore College for two years, while studying piano at the Leefson Conservatory of Music in Philadelphia. He earned a master's degree from Johns Hopkins University, and a Ph.D. degree from Princeton University in 1939. His doctoral dissertation, On the LusternickSchnirelmann Category, was directed by Solomon Lefschetz. (In later years he disclaimed all knowledge of Lusternik–Schnirelmann category, and certainly never published on the subject again.) He directed 21 doctoral dissertations, including those of John Milnor, John Stallings, Francisco GonzálezAcuña, Guillermo TorresDiaz and Barry Mazur.
He was an Invited Speaker at the International Congress of Mathematicians held in 1950 in Cambridge, Massachusetts.^{[1]} His mathematical contributions include Fox ncoloring of knots, the FoxArtin arc, and the free differential calculus. He also identified the compactopen topology on function spaces as being particularly appropriate for homotopy theory.
Aside from his strictly mathematical contributions, he was responsible for introducing several basic phrases to knot theory: the phrases slice knot, ribbon knot, and Seifert circle all appear in print for the first time under his name, and he also popularized (if he did not introduce) the phrase Seifert surface.
He popularized the playing of the game of Go at both Princeton and the Institute for Advanced Study.
Selected publications
 Introduction to Knot Theory, Richard H. Crowell and Ralph H. Fox, Reprint of the 1963 original, Graduate Texts in Mathematics, No. 57, SpringerVerlag, New YorkHeidelberg, 1977. ISBN 0387902724^{[2]}
 "A quick trip through knot theory", in: M. K. Fort (Ed.), Topology of 3Manifolds and Related Topics, PrenticeHall, New Jersey, 1961, pp. 120–167. MR 0140099
 "Metacyclic invariants of knots and links". Canadian Journal of Mathematics. 22: 193–201. 1970. MR 0261584.
 "Rolling". Bulletin of the American Mathematical Society. 72, Part 1: 162–164. 1966. doi:10.1090/s000299041966114672. MR 0184221.
 with N. Smythe: "An ideal class invariant of knots". Bulletin of the American Mathematical Society. 15: 707–709. 1964. doi:10.1090/s00029939196401655162. MR 0165516.
 "On topologies for function spaces". Bulletin of the American Mathematical Society. 51: 429–432. 1945. doi:10.1090/s000299041945083700. MR 0012224
 with W. A. Blankinship: "Remarks on certain pathological open subsets of 3space and their fundamental groups". Proc. Amer. Math. Soc. 1: 618–624. 1950. doi:10.1090/s00029939195000421208. MR 0042120.
 "Torus Homotopy Groups". Proceedings of the National Academy of Sciences of the United States of America. 31 (2): 71–74. February 1945. doi:10.1073/pnas.31.2.71. PMC 1078755. PMID 16588687.
 "On fibre spaces. I". Bulletin of the American Mathematical Society. 49: 555–557. 1943. doi:10.1090/s000299041943079694. MR 0008702.
 "On fibre spaces. II". Bulletin of the American Mathematical Society. 49: 733–735. 1943. doi:10.1090/s000299041943080159. MR 0009109.
 "A characterization of absolute neighborhood retracts". Bulletin of the American Mathematical Society. 48: 271–275. 1942. doi:10.1090/s00029904194207652x. MR 0006508.
 "On Homotopy and Extension of Mappings". Proceedings of the National Academy of Sciences of the United States of America. 26 (1): 26–28. 15 January 1940. doi:10.1073/pnas.26.1.26. PMC 1078000. PMID 16577957.
 Fox, Ralph; Neuwirth, Lee (1962). "The braid groups". Mathematica Scandinavica. 10: 119–126. doi:10.7146/math.scand.a10518. MR 0150755.
References
 ^ Fox, Ralph H. (1950). "Recent developments of knot theory at Princeton" (PDF). Proceedings of the International Congress of Mathematicians, Cambridge, Massachusetts, U.S.A., August 30–September 6, 1950. 2. pp. 453–458.
 ^ Neuwirth, Lee P. (1964). "Review: Introduction to knot theory by R. H. Crowell and R. H. Fox" (PDF). Bulletin of the American Mathematical Society. 70 (2): 235–238. doi:10.1090/s00029904196411096x.
External links
 O'Connor, John J.; Robertson, Edmund F., "Ralph Fox", MacTutor History of Mathematics archive, University of St Andrews.
 Ralph Fox at the Mathematics Genealogy Project
 [1] Jozef H. Przytycki, Notes to the early history of the Knot Theory in Japan, 2001.