Knot Theory and Khovanov Homology

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Knot Theory and Khovanov Homology Juan Ariel Ortiz Navarro juorna@gmail.com Departamento de Ciencias Matemáticas Universidad de Puerto Rico - Mayagüez JAON-SACNAS, Dallas, TX p.1/15

Knot Theory Motivated in 1880 by chemists JAON-SACNAS, Dallas, TX p.2/15

Knot Theory Motivated in 1880 by chemists Embedding of a collection of S 1 into R 3 JAON-SACNAS, Dallas, TX p.2/15

Knot Theory Motivated in 1880 by chemists Embedding of a collection of S 1 into R 3 JAON-SACNAS, Dallas, TX p.2/15

Knot Theory Motivated in 1880 by chemists Embedding of a collection of S 1 into R 3 JAON-SACNAS, Dallas, TX p.2/15

Knot Theory Motivated in 1880 by chemists Embedding of a collection of S 1 into R 3 JAON-SACNAS, Dallas, TX p.2/15

Knot Theory Motivated in 1880 by chemists Embedding of a collection of S 1 into R 3 A knot or link has many different diagrams JAON-SACNAS, Dallas, TX p.2/15

Knot Theory Motivated in 1880 by chemists Embedding of a collection of S 1 into R 3 A knot or link has many different diagrams JAON-SACNAS, Dallas, TX p.2/15

Knot Theory Motivated in 1880 by chemists Embedding of a collection of S 1 into R 3 A knot or link has many different diagrams JAON-SACNAS, Dallas, TX p.2/15

Knot Theory Reidemeister (20 s): Two diagrams represent the same link if and only if there is a series of Reidemeister moves from one to the other JAON-SACNAS, Dallas, TX p.3/15

Knot Theory Reidemeister (20 s): Two diagrams represent the same link if and only if there is a series of Reidemeister moves from one to the other Type I Type II Type III JAON-SACNAS, Dallas, TX p.3/15

Knot Theory Reidemeister (20 s): Two diagrams represent the same link if and only if there is a series of Reidemeister moves from one to the other Link Invariant: Property unchanged by these moves Type I Type II Type III JAON-SACNAS, Dallas, TX p.3/15

Link Invariants JAON-SACNAS, Dallas, TX p.4/15

Link Invariants Algebraic topology: JAON-SACNAS, Dallas, TX p.4/15

Link Invariants Algebraic topology: invariants are algebraic objects JAON-SACNAS, Dallas, TX p.4/15

Link Invariants Algebraic topology: invariants are algebraic objects Numbers JAON-SACNAS, Dallas, TX p.4/15

Link Invariants Algebraic topology: invariants are algebraic objects Numbers unknotting number JAON-SACNAS, Dallas, TX p.4/15

Link Invariants Algebraic topology: invariants are algebraic objects Numbers unknotting number stick number JAON-SACNAS, Dallas, TX p.4/15

Link Invariants Algebraic topology: invariants are algebraic objects Numbers unknotting number stick number Groups JAON-SACNAS, Dallas, TX p.4/15

Link Invariants Algebraic topology: invariants are algebraic objects Numbers unknotting number stick number Groups Polynomials JAON-SACNAS, Dallas, TX p.4/15

Link Invariants Algebraic topology: invariants are algebraic objects Numbers unknotting number stick number Groups Polynomials Alexander JAON-SACNAS, Dallas, TX p.4/15

Link Invariants Algebraic topology: invariants are algebraic objects Numbers unknotting number stick number Groups Polynomials Alexander Jones JAON-SACNAS, Dallas, TX p.4/15

Link Invariants Algebraic topology: invariants are algebraic objects Numbers unknotting number stick number Groups Polynomials Alexander Jones Kauffman s Bracket JAON-SACNAS, Dallas, TX p.4/15

Kauffman s bracket The Kauffman s bracket is defined by: JAON-SACNAS, Dallas, TX p.5/15

Kauffman s bracket The Kauffman s bracket is defined by: < >= 1 < unknot L >= (q + q 1 ) < L > < D >= < D + > q < D > JAON-SACNAS, Dallas, TX p.5/15

Kauffman s bracket The Kauffman s bracket is defined by: < >= 1 < unknot L >= (q + q 1 ) < L > < D >= < D + > q < D > D D+ D JAON-SACNAS, Dallas, TX p.5/15

Kauffman s Bracket for the Hopf Link JAON-SACNAS, Dallas, TX p.6/15

Kauffman s Bracket for the Hopf Link q JAON-SACNAS, Dallas, TX p.6/15

Kauffman s Bracket for the Hopf Link q q q q JAON-SACNAS, Dallas, TX p.6/15

Kauffman s Bracket for the Hopf Link q q q q 1 q q q q 1 q q q 1 JAON-SACNAS, Dallas, TX p.6/15

Kauffman s Bracket for the Hopf Link q q q q 1 q q q q 1 q q q 1 3 q q 1 JAON-SACNAS, Dallas, TX p.6/15

Kauffman s Bracket for the Hopf Link q q q q 1 q q q q 1 q q q 1 3 q q 1 1 3 q q q q 1 JAON-SACNAS, Dallas, TX p.6/15

Kauffman s Bracket for the Hopf Link q q q q 1 q q q q 1 q q q 1 3 q q 1 q 1 3 q q 1 q 2 4 q 2 q q 1 JAON-SACNAS, Dallas, TX p.6/15

Jones Polynomial Oriented knots or links have two types of crossings: JAON-SACNAS, Dallas, TX p.7/15

Jones Polynomial Oriented knots or links have two types of crossings: positive JAON-SACNAS, Dallas, TX p.7/15

Jones Polynomial Oriented knots or links have two types of crossings: positive negative JAON-SACNAS, Dallas, TX p.7/15

Jones Polynomial Oriented knots or links have two types of crossings: positive negative Let n + be the number of positive crossings and n be the number of negative crossings. JAON-SACNAS, Dallas, TX p.7/15

Jones Polynomial Oriented knots or links have two types of crossings: positive negative Let n + be the number of positive crossings and n be the number of negative crossings. Then the Jones Polynomial J L (q) of the link L is: J L (q) = 1 n q n + 2n < L > JAON-SACNAS, Dallas, TX p.7/15

Jones Polynomial for the Hopf Link JAON-SACNAS, Dallas, TX p.8/15

Jones Polynomial for the Hopf Link In this diagram n + = 2 and n = 0. Then the Jones Polynomial for the Hopf Link is: JAON-SACNAS, Dallas, TX p.8/15

Jones Polynomial for the Hopf Link In this diagram n + = 2 and n = 0. Then the Jones Polynomial for the Hopf Link is: J hopf (q) = ( 1) 0 q 2 (q 2 + 1 + q 2 + q 4 ) = 1 + q 2 + q 4 + q 6 JAON-SACNAS, Dallas, TX p.8/15

Khovanov Homology M. Khovanov (1999) JAON-SACNAS, Dallas, TX p.9/15

Khovanov Homology M. Khovanov (1999) Assigns homology groups to a diagram of a knot (or link) JAON-SACNAS, Dallas, TX p.9/15

Khovanov Homology M. Khovanov (1999) Assigns homology groups to a diagram of a knot (or link) Knots (and links) invariant JAON-SACNAS, Dallas, TX p.9/15

Khovanov Homology M. Khovanov (1999) Assigns homology groups to a diagram of a knot (or link) Knots (and links) invariant Categorification of the Jones polynomial JAON-SACNAS, Dallas, TX p.9/15

Khovanov Homology M. Khovanov (1999) Assigns homology groups to a diagram of a knot (or link) Knots (and links) invariant Categorification of the Jones polynomial Uses the algebra A =< 1,x > over a field F with the following operations: A A m A, m(1 b) = m(b 1) = b, m(x x) = 0 A A A, (1) = 1 x + x 1, (x) = x x JAON-SACNAS, Dallas, TX p.9/15

Khovanov Complex Order the vertices: JAON-SACNAS, Dallas, TX p.10/15

Khovanov Complex Order the vertices: A B JAON-SACNAS, Dallas, TX p.10/15

Khovanov Complex Order the vertices: A Two ways of smoothing a crossing: B JAON-SACNAS, Dallas, TX p.10/15

Khovanov Complex Order the vertices: A Two ways of smoothing a crossing: B 0-smoothing 1-smoothing JAON-SACNAS, Dallas, TX p.10/15

Khovanov Complex Order the vertices: A Two ways of smoothing a crossing: B 0-smoothing 1-smoothing For each vertex α of the n-dimensional cube it assigns a particular way of smoothing the knot: α state α JAON-SACNAS, Dallas, TX p.10/15

Khovanov Complex A B JAON-SACNAS, Dallas, TX p.11/15

Khovanov Complex A B vertex (0, 0) JAON-SACNAS, Dallas, TX p.11/15

Khovanov Complex A (0,0) B vertex (0, 0) JAON-SACNAS, Dallas, TX p.11/15

Khovanov Complex A (0,0) B vertex (0, 0) To each state α it assigns the vector space [L] α = #circles i=1 A JAON-SACNAS, Dallas, TX p.11/15

Khovanov Complex A (0,0) B vertex (0, 0) To each state α it assigns the vector space [L] α = #circles i=1 A The (0, 0)-state is assigned the vector space: A A JAON-SACNAS, Dallas, TX p.11/15

Khovanov Complex A (0,0) B vertex (0, 0) To each state α it assigns the vector space [L] α = #circles i=1 A The (0, 0)-state is assigned the vector space: A A Let C i = α =i [L] α to get the complex with -operator defined by the operations on the algebra A JAON-SACNAS, Dallas, TX p.11/15

Complex for the Hopf link (1,0) (0,0) m A (1,1) A x A m (0,1) A x A C 0 (L){0}{2} 0 A 1 1 C (L){1}{2} C 2 (L){2}{2} JAON-SACNAS, Dallas, TX p.12/15

Khovanov Homology JAON-SACNAS, Dallas, TX p.13/15

Khovanov Homology C(L) is a complex, i.e., i i 1 = 0 for every i. JAON-SACNAS, Dallas, TX p.13/15

Khovanov Homology C(L) is a complex, i.e., i i 1 = 0 for every i. Let Z i = ker i and B i = Im i, JAON-SACNAS, Dallas, TX p.13/15

Khovanov Homology C(L) is a complex, i.e., i i 1 = 0 for every i. Let Z i = ker i and B i = Im i, then H i = Z i /B i 1 JAON-SACNAS, Dallas, TX p.13/15

Khovanov Homology C(L) is a complex, i.e., i i 1 = 0 for every i. Let Z i = ker i and B i = Im i, then H i = Z i /B i 1 The homology H i (L) of the complex C(L) is an invariant for the link L and it is called the Khovanov homology of L. JAON-SACNAS, Dallas, TX p.13/15

Khovanov Homology C(L) is a complex, i.e., i i 1 = 0 for every i. Let Z i = ker i and B i = Im i, then H i = Z i /B i 1 The homology H i (L) of the complex C(L) is an invariant for the link L and it is called the Khovanov homology of L. From this groups one can obtain the Jones polynomial. JAON-SACNAS, Dallas, TX p.13/15

Khovanov Homology C(L) is a complex, i.e., i i 1 = 0 for every i. Let Z i = ker i and B i = Im i, then H i = Z i /B i 1 The homology H i (L) of the complex C(L) is an invariant for the link L and it is called the Khovanov homology of L. From this groups one can obtain the Jones polynomial. In general, the Khovanov homology is stronger (invariant) than the Jones polynomial. JAON-SACNAS, Dallas, TX p.13/15

KH for the Hopf Link The complex: 0 C 0 0 C 1 1 C 2 0 JAON-SACNAS, Dallas, TX p.14/15

KH for the Hopf Link The complex: 0 C 0 0 C 1 1 C 2 0 1 1 0 (1, 1) x 1 0 (x,x) 1 x 0 (x,x) x x 0 (0, 0) JAON-SACNAS, Dallas, TX p.14/15

KH for the Hopf Link The complex: 0 C 0 0 C 1 1 C 2 0 1 1 0 (1, 1) x 1 0 (x,x) 1 x 0 (x,x) x x 0 (0, 0) (1, 0) 1 1 x x 1 (x, 0) 1 x x (0, 1) 1 1 x + x 1 (0,x) 1 x x JAON-SACNAS, Dallas, TX p.14/15

KH for the Hopf Link The complex: 0 C 0 0 C 1 1 C 2 0 1 1 0 (1, 1) x 1 0 (x,x) 1 x 0 (x,x) x x 0 (0, 0) Z 0 =< 1 x x 1,x x > (1, 0) 1 1 x x 1 (x, 0) 1 x x (0, 1) 1 1 x + x 1 (0,x) 1 x x JAON-SACNAS, Dallas, TX p.14/15

KH for the Hopf Link The complex: 0 C 0 0 C 1 1 C 2 0 1 1 0 (1, 1) (1, 0) 1 1 x x 1 x 1 0 (x,x) 1 x 0 (x,x) x x 0 (0, 0) Z 0 =< 1 x x 1,x x > B 0 =< (1, 1), (x,x) > (x, 0) 1 x x (0, 1) 1 1 x + x 1 (0,x) 1 x x JAON-SACNAS, Dallas, TX p.14/15

KH for the Hopf Link The complex: 0 C 0 0 C 1 1 C 2 0 1 1 0 (1, 1) (1, 0) 1 1 x x 1 x 1 0 (x,x) 1 x 0 (x,x) x x 0 (0, 0) Z 0 =< 1 x x 1,x x > B 0 =< (1, 1), (x,x) > (x, 0) 1 x x (0, 1) 1 1 x + x 1 (0,x) 1 x x Z 1 =< (1, 1), (x,x) > JAON-SACNAS, Dallas, TX p.14/15

KH for the Hopf Link The complex: 0 C 0 0 C 1 1 C 2 0 1 1 0 (1, 1) (1, 0) 1 1 x x 1 x 1 0 (x,x) 1 x 0 (x,x) x x 0 (0, 0) Z 0 =< 1 x x 1,x x > B 0 =< (1, 1), (x,x) > (x, 0) 1 x x (0, 1) 1 1 x + x 1 (0,x) 1 x x Z 1 =< (1, 1), (x,x) > B 1 =< 1 x + x 1,x x JAON-SACNAS, Dallas, TX p.14/15

KH for the Hopf Link The complex: 0 C 0 0 C 1 1 C 2 0 1 1 0 (1, 1) (1, 0) 1 1 x x 1 x 1 0 (x,x) 1 x 0 (x,x) x x 0 (0, 0) Z 0 =< 1 x x 1,x x > B 0 =< (1, 1), (x,x) > H 0 =< x 1 1 x,x x > (x, 0) 1 x x (0, 1) 1 1 x + x 1 (0,x) 1 x x Z 1 =< (1, 1), (x,x) > B 1 =< 1 x + x 1,x x JAON-SACNAS, Dallas, TX p.14/15

KH for the Hopf Link The complex: 0 C 0 0 C 1 1 C 2 0 1 1 0 (1, 1) (1, 0) 1 1 x x 1 x 1 0 (x,x) 1 x 0 (x,x) (x, 0) 1 x x (0, 1) 1 1 x + x 1 x x 0 (0, 0) Z 0 =< 1 x x 1,x x > (0,x) 1 x x Z 1 =< (1, 1), (x,x) > B 0 =< (1, 1), (x,x) > B 1 =< 1 x + x 1,x x H 0 =< x 1 1 x,x x >, H 1 = 0 JAON-SACNAS, Dallas, TX p.14/15

KH for the Hopf Link The complex: 0 C 0 0 C 1 1 C 2 0 1 1 0 (1, 1) (1, 0) 1 1 x x 1 x 1 0 (x,x) 1 x 0 (x,x) (x, 0) 1 x x (0, 1) 1 1 x + x 1 x x 0 (0, 0) Z 0 =< 1 x x 1,x x > (0,x) 1 x x Z 1 =< (1, 1), (x,x) > B 0 =< (1, 1), (x,x) > B 1 =< 1 x + x 1,x x H 0 =< x 1 1 x,x x >, H 1 = 0, and H 2 =< 1 1, 1 x x 1 > JAON-SACNAS, Dallas, TX p.14/15

REFERENCES C. Adams, The Knot Book D. Bar-Natan, On Khovanov s categorification of the Jones Polynomial, Algebr. and Geom. Topology, 2002 M. Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (3), 1999 A. Shumakovitch, KhoHo version 0.9.3.5, http://www.geometrie.ch/khoho JAON-SACNAS, Dallas, TX p.15/15

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