Tranverse invariants & Kh-type Homolo- gies Invariants for transverse knots from Khovanov-type homologies Contact & links Kh-type homolo- gies Carlo Collari Invariants Universit` a degli studi di Firenze
Tranverse invariants & Kh-type Homolo- gies 1 Contact structures, links and braids Contact & links Kh-type homolo- gies Invariants 2 Khovanov-Type homologies 3 Transverse invariants and Khovanov-type homologies
Contact manifolds Tranverse invariants & Kh-type Homolo- gies Definition Let M be an odd-dimensional manifold. A contact structure ξ (on M ) is a Contact & totally non-integrable hyperplane field. links Kh-type homolo- gies Invariants The symmetric structure on R 3 is given by ξ sym = Ker ( dz + xdy − ydx ); Figure: source Wikipedia
Contact manifolds Tranverse invariants & Kh-type Homolo- gies Definition Let M be an odd-dimensional manifold. A contact structure ξ (on M ) is a Contact & totally non-integrable hyperplane field. links Kh-type homolo- gies Invariants The symmetric structure on R 3 is given by ξ sym = Ker ( dz + xdy − ydx ); Figure: source Wikipedia
Links in contact manifolds Tranverse invariants & Kh-type Homolo- gies Definition Let ( M 3 , ξ ) be a contact (3-)manifold. A (smooth) link in M is called Contact & links 1 Legendrian if it is everywhere tangent to the contact structure; Kh-type homolo- 2 transverse if it is everywhere transverse to the contact structure. gies Invariants Figure: Interaction between a contact plane, and a Legendrian (left), resp. transverse (right), link.
Links in contact manifolds Tranverse invariants & Kh-type Homolo- gies Definition Let ( M 3 , ξ ) be a contact (3-)manifold. A (smooth) link in M is called Contact & links 1 Legendrian if it is everywhere tangent to the contact structure; Kh-type homolo- 2 transverse if it is everywhere transverse to the contact structure. gies Invariants Figure: Interaction between a contact plane, and a Legendrian (left), resp. transverse (right), link.
Links in contact manifolds Tranverse invariants & Kh-type Homolo- gies Definition Let ( M 3 , ξ ) be a contact (3-)manifold. A (smooth) link in M is called Contact & links 1 Legendrian if it is everywhere tangent to the contact structure; Kh-type homolo- 2 transverse if it is everywhere transverse to the contact structure. gies Invariants Figure: Interaction between a contact plane, and a Legendrian (left), resp. transverse (right), link.
Links in contact manifolds Tranverse invariants & Kh-type Homolo- gies Definition Let ( M 3 , ξ ) be a contact (3-)manifold. A (smooth) link in M is called Contact & links 1 Legendrian if it is everywhere tangent to the contact structure; Kh-type homolo- 2 transverse if it is everywhere transverse to the contact structure. gies Invariants Figure: Interaction between a contact plane, and a Legendrian (left), resp. transverse (right), link.
Braids Tranverse invariants 1 1 . . . . & . . . . Kh-type Definition . . . . Homolo- k - 1 k - 1 gies The braid group on i -strands, k k denoted by B i , is the group k + 1 k + 1 Contact & generated by σ j , for j ∈ { 1 , ..., i } , links k+2 k+2 and subject to the following . . . . . . . . Kh-type . . . . homolo- relations: i i gies σ − 1 σ k ∈ B i ∈ B i k Invariants σ k σ j = σ j σ k , | k − j | > 1 , Figure: Generators of the Braid group σ k +1 σ k σ k +1 = σ k σ k +1 σ k . B i . � . . . . . . . . i strands T T ′ . . . . Figure: Operation in the braid group B i .
Braids Tranverse invariants 1 1 . . . . & . . . . Kh-type Definition . . . . Homolo- k - 1 k - 1 gies The braid group on i -strands, k k denoted by B i , is the group k + 1 k + 1 Contact & generated by σ j , for j ∈ { 1 , ..., i } , links k+2 k+2 and subject to the following . . . . . . . . Kh-type . . . . homolo- relations: i i gies σ − 1 σ k ∈ B i ∈ B i k Invariants σ k σ j = σ j σ k , | k − j | > 1 , Figure: Generators of the Braid group σ k +1 σ k σ k +1 = σ k σ k +1 σ k . B i . � . . . . . . . . i strands T T ′ . . . . Figure: Operation in the braid group B i .
Braids Tranverse invariants 1 1 . . . . & . . . . Kh-type Definition . . . . Homolo- k - 1 k - 1 gies The braid group on i -strands, k k denoted by B i , is the group k + 1 k + 1 Contact & generated by σ j , for j ∈ { 1 , ..., i } , links k+2 k+2 and subject to the following . . . . . . . . Kh-type . . . . homolo- relations: i i gies σ − 1 σ k ∈ B i ∈ B i k Invariants σ k σ j = σ j σ k , | k − j | > 1 , Figure: Generators of the Braid group σ k +1 σ k σ k +1 = σ k σ k +1 σ k . B i . � . . . . . . . . i strands T T ′ . . . . Figure: Operation in the braid group B i .
Operations on braids Tranverse invariants & Definition Kh-type Homolo- . . gies . . A braid T is an element of any B i , T . . and the integer i = i ( T ) will be called braid index of T . Contact & links Figure: Negative stabilization of the Kh-type braid T homolo- gies Invariants . . . . T . . . . . . T . . Figure: Positive stabilization of the braid T Figure: Alexander closure of the braid T
Operations on braids Tranverse invariants & Definition Kh-type Homolo- . . gies . . A braid T is an element of any B i , T . . and the integer i = i ( T ) will be called braid index of T . Contact & links Figure: Negative stabilization of the Kh-type braid T homolo- gies Invariants . . . . T . . . . . . T . . Figure: Positive stabilization of the braid T Figure: Alexander closure of the braid T
Operations on braids Tranverse invariants & Definition Kh-type Homolo- . . gies . . A braid T is an element of any B i , T . . and the integer i = i ( T ) will be called braid index of T . Contact & links Figure: Negative stabilization of the Kh-type braid T homolo- gies Invariants . . . . T . . . . . . T . . Figure: Positive stabilization of the braid T Figure: Alexander closure of the braid T
Operations on braids Tranverse invariants & Definition Kh-type Homolo- . . gies . . A braid T is an element of any B i , T . . and the integer i = i ( T ) will be called braid index of T . Contact & links Figure: Negative stabilization of the Kh-type braid T homolo- gies Invariants . . . . T . . . . . . T . . Figure: Positive stabilization of the braid T Figure: Alexander closure of the braid T
Transverse links and Braids Tranverse invariants & Kh-type Homolo- gies Theorem (Bennequin, ’83) Contact & ⊙ links Any transverse link in ( R 3 , ξ sym ) is Kh-type . . transversely isotopic to the homolo- . . T . . gies Alexander closure of a braid. Invariants Theorem (Orevkov and Shevchishin, Wrinkle, ’03) Two braids represent the same transverse link type if, and only if, they are related by a finite sequence of conjugations in the braid group, positive stabilizations and positive destabilizations.
Transverse links and Braids Tranverse invariants & Kh-type Homolo- gies Theorem (Bennequin, ’83) Contact & ⊙ links Any transverse link in ( R 3 , ξ sym ) is Kh-type . . transversely isotopic to the homolo- . . T . . gies Alexander closure of a braid. Invariants Theorem (Orevkov and Shevchishin, Wrinkle, ’03) Two braids represent the same transverse link type if, and only if, they are related by a finite sequence of conjugations in the braid group, positive stabilizations and positive destabilizations.
Transverse links and Braids Tranverse invariants & Kh-type Homolo- gies Theorem (Bennequin, ’83) Contact & ⊙ links Any transverse link in ( R 3 , ξ sym ) is Kh-type . . transversely isotopic to the homolo- . . T . . gies Alexander closure of a braid. Invariants Theorem (Orevkov and Shevchishin, Wrinkle, ’03) Two braids represent the same transverse link type if, and only if, they are related by a finite sequence of conjugations in the braid group, positive stabilizations and positive destabilizations.
Classical invariants Tranverse invariants & Kh-type Homolo- gies There are two classical invariants for transverse links: 1 the link-type ; Contact & links 2 the self-linking number ; Kh-type homolo- The latter could be defined, in the case of a braid T , as gies Invariants sl ( T ) = n + ( T ) − n − ( T ) − i ( T ) . Any invariant which is strictly more powerful than sl and the link-type is called effective . A family of transverse link whose elements are told apart one from the other by the two classical invariants is called simple (e.g. the unknot, torus knots, figure eight).
Classical invariants Tranverse invariants & Kh-type Homolo- gies There are two classical invariants for transverse links: 1 the link-type ; Contact & links 2 the self-linking number ; Kh-type homolo- The latter could be defined, in the case of a braid T , as gies Invariants sl ( T ) = n + ( T ) − n − ( T ) − i ( T ) . Any invariant which is strictly more powerful than sl and the link-type is called effective . A family of transverse link whose elements are told apart one from the other by the two classical invariants is called simple (e.g. the unknot, torus knots, figure eight).
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