Selected advances in knot theory |
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Years (started) |
Idea |
People |
Introductory reading |
1870's |
Knot tables and diagrammatic invariants |
P. G. Tait, T. P. Kirkman, C. N. Little |
"Knot Theory's Odd Origins" by D.Silver, History of math website. |
1914 |
Trefoils are not equivalent by knot group |
M. Dehn |
Burde-Zieschang |
1923 |
Every link is a closed braid |
E. Artin, J. W. Alexander |
Rolfsen |
1928 |
Alexander polynomial |
J. W. Alexander |
Rolfsen |
1932 |
Reidemeister moves |
K. Reidemeister |
Any book about knot theory |
1934 |
Seifert surfaces and matrices |
H. Seifert |
Rolfsen |
1950's |
Wild knots, Fox calculus, algebraic topology |
R. H. Fox, J. Milnor, et al |
"A quick trip through knot theory" by R.H. Fox |
1954 |
PL and smooth knot types are equivalent |
E. Moise |
Burde-Zieschang |
1960's |
Fibered knots |
J. Stallings, J. Milnor, et al |
Rolfsen, Burde-Zieschang |
1961 |
Unknotting problem is solvable |
W. Haken |
|
1962 |
Any closed orientable 3-manifold is given by a surgery on a link in 3-sphere |
W. Lickorish, A. Wallace |
Lickorish, Rolfsen |
1964 |
Non-invertible knots exist |
H. Trotter |
|
1967 |
Rational tangles, Conway polynomial |
J. H. Conway |
Cromwell |
1970's |
Concordance group (slice knots), signature |
J. Milnor, K. Murasugi, T. Cochran, et al |
"A survey of classical knot concordance" by C.Livingston |
1970's |
Cyclic branched covers |
H. Hilden, J. Montesinos |
Rolfsen, Prasolov-Sossinsky |
1976 |
Kirby calculus |
R. Kirby |
Lickorish, Rolfsen |
1980's |
Geometrization of 3-manifolds |
W. Thurston |
"Low-dimensional geometry" by F.Bonahon |
1980's |
Essential surfaces in knot complements, exceptional surgeries, boundary slopes |
W. Thurston, M. Culler, P. Shalen, C. Gordon, et al |
"Dehn surgery on knots in the 3-sphere" by J.Luecke |
1984 |
Jones polynomial |
V. Jones |
Lickorish |
1985 |
HOMFLYPT polynomial |
many |
Lickorish |
1987 |
Kauffman bracket and state sums |
L. Kauffman |
Lickorish |
1980's |
Tait's conjectures for alternating links proved |
W. Menasco, M. Thistlethwaite, K. Murasugi, L. Kauffman |
|
1989 |
Knots are determined by their complements |
C. Gordon and J. Luecke |
|
1989 |
Physical idea for Jones polynomial of knots in 3-manifolds via path integrals |
E. Witten |
|
1990's |
Quantum invariants of links in 3-manifolds |
N. Reshetikhin, V. Turaev, O. Viro, W. Lickorish |
Lickorish |
1990's |
Skein algebras & modules, Temperley-Lieb algebra |
J. Przytycki, L. Kauffman, C. Frohman, G. Masbaum |
Prasolov-Sossinsky |
1990's |
Finite type invariants |
V. Vassiliev, D. Bar-Natan, M. Kontsevich |
Chmutov-Duzhin-Mostovoy |
1990's |
Colored Jones polynomials, Melvin-Morton Conj |
D. Bar-Natan, S. Garoufalidis, et al |
Chmutov-Duzhin-Mostovoy |
1990's |
Braid groups are linear |
J. Birman, S. Bigelow, R. Lawrence, D. Krammer |
"Braids: A survey" by Birman and Brendle |
1990's |
Legendrian and transversal knots |
J. Etnyre, L. Ng, et al |
"Legendrian and transversal knots" by J.Etnyre |
1994 |
A-polynomial |
M. Culler, P. Shalen, N. Dunfield, et al |
"Non-triviality of the A-polynomial for knots in S^3" by Dunfield-Garoufalidis |
1990's |
Twisted Alexander polynomials |
X.S. Lin, P. Kirk, C. Livingston, S. Friedl |
"A survey of twisted Alexander polynomials" by Friedl-Vidussi |
1990's |
Knotscape, SnapPea, Mathematica |
J. Weeks, M. Thistlethwaite, D. Bar-Natan |
http://www.math.uiuc.edu/~nmd/computop/ |
1990's |
Computational complexity problems |
J. Hass, J. Lagarias, G. Kuperberg |
"Knottedness is in NP modulo GRH" by G.Kuperberg |
2000's |
Khovanov homology, categorification |
M. Khovanov, L. Rozansky, et al |
"Five lectures on Khovanov homology" by P.Turner |
2000's |
Heegaard-Floer homology |
P. Ozsvath, Z. Szabo, et al |
"An introduction to knot Floer homology" by C.Manolescu |
2000's |
Quantum topology, AJ Conjecture, q-holonomicity |
S. Garoufalidis, Thang Le, T.Ohtsuki, et al |
"Is the Jones polynomial of a knot really a polynomial" by Garoufalidis-Le |
2000's |
Volume Conjecture |
R. Kashaev, H. Murakami, S. Gukov |
"An introduction to the Volume Conjecture" by H.Murakami |
2000's |
Geometric bounds from link diagrams |
M. Lackenby, I. Agol, D. Thurston, D. Futer, J. Purcell |
Bonahon, Futer-Kalfagianni-Purcell |
2000's |
Knot Atlas, KnotInfo, SnapPy |
D. Bar-Natan, C. Livingston, N. Dunfield, et al |
http://www.math.uiuc.edu/~nmd/computop/ |