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

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/