Selected advances in knot theory 

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 
BurdeZieschang 
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 
BurdeZieschang 
1960's 
Fibered knots 
J. Stallings, J. Milnor, et al 
Rolfsen, BurdeZieschang 
1961 
Unknotting problem is solvable 
W. Haken 

1962 
Any closed orientable 3manifold is given by a surgery on a link in 3sphere 
W. Lickorish, A. Wallace 
Lickorish, Rolfsen 
1964 
Noninvertible 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, PrasolovSossinsky 
1976 
Kirby calculus 
R. Kirby 
Lickorish, Rolfsen 
1980's 
Geometrization of 3manifolds 
W. Thurston 
"Lowdimensional 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 3sphere" 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 3manifolds via path integrals 
E. Witten 

1990's 
Quantum invariants of links in 3manifolds 
N. Reshetikhin, V. Turaev, O. Viro, W. Lickorish 
Lickorish 
1990's 
Skein algebras & modules, TemperleyLieb algebra 
J. Przytycki, L. Kauffman, C. Frohman, G. Masbaum 
PrasolovSossinsky 
1990's 
Finite type invariants 
V. Vassiliev, D. BarNatan, M. Kontsevich 
ChmutovDuzhinMostovoy 
1990's 
Colored Jones polynomials, MelvinMorton Conj 
D. BarNatan, S. Garoufalidis, et al 
ChmutovDuzhinMostovoy 
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 
Apolynomial 
M. Culler, P. Shalen, N. Dunfield, et al 
"Nontriviality of the Apolynomial for knots in S^3" by DunfieldGaroufalidis 
1990's 
Twisted Alexander polynomials 
X.S. Lin, P. Kirk, C. Livingston, S. Friedl 
"A survey of twisted Alexander polynomials" by FriedlVidussi 
1990's 
Knotscape, SnapPea, Mathematica 
J. Weeks, M. Thistlethwaite, D. BarNatan 
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 
HeegaardFloer homology 
P. Ozsvath, Z. Szabo, et al 
"An introduction to knot Floer homology" by C.Manolescu 
2000's 
Quantum topology, AJ Conjecture, qholonomicity 
S. Garoufalidis, Thang Le, T.Ohtsuki, et al 
"Is the Jones polynomial of a knot really a polynomial" by GaroufalidisLe 
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, FuterKalfagianniPurcell 
2000's 
Knot Atlas, KnotInfo, SnapPy 
D. BarNatan, C. Livingston, N. Dunfield, et al 
http://www.math.uiuc.edu/~nmd/computop/ 