Preliminaries Results Appendices Quasi-invariants of 2-knots and quantum integrable systems Dmitry Talalaev MSU, ITEP May,2015, GGI, Firenze D. Talalaev Quasi-invariants of 2-knots and quantum integrable systems
Preliminaries Results Appendices References This talk is based on the papers: I. Korepanov, G. Sharygin, D.T: ”Cohomologies of n -simplex relations”, arXiv:1409.3127 D.T. ”Zamolodchikov tetrahedral equation and higher Hamiltonians of 2d quantum integrable systems”, arXiv:1505.06579 ”Cohomology of the tetrahedral complex and quasi-invariants of 2-knots.” in progress D. Talalaev Quasi-invariants of 2-knots and quantum integrable systems
Preliminaries Tetrahedral equation Results 2-knot diagram Appendices Preliminaries 1 Tetrahedral equation 2-knot diagram Results 2 Quasi-invariants Regular lattices and 2d quantum integrability Summary and perspectives Appendices 3 Tetrahedral complex Quandle cohomology and 2-knot invariants D. Talalaev Quasi-invariants of 2-knots and quantum integrable systems
Preliminaries Tetrahedral equation Results 2-knot diagram Appendices A. Zamolodchikov [1981] Vector version Let Φ ∈ End ( V ⊗ 3 ) , where V - (f.d) vector space. The tetrahedral equation takes the form Φ 123 Φ 145 Φ 246 Φ 356 = Φ 356 Φ 246 Φ 145 Φ 123 where both sides are linear operators in V ⊗ 6 and Φ ijk represents the operator acting in components i , j , k as Φ and trivially in the others. Figure : Tetrahedral equation D. Talalaev Quasi-invariants of 2-knots and quantum integrable systems
Preliminaries Tetrahedral equation Results 2-knot diagram Appendices Set-theoretic version Let X be a (f) set. We say that a map R X × X × X − → X × X × X , satisfy the s.t. tetrahedral equation if R 123 ◦ R 145 ◦ R 246 ◦ R 356 = R 356 ◦ R 246 ◦ R 145 ◦ R 123 where both sides are maps of the Cartesian power X × 6 and the subscripts correspond to components of X . For example R 356 ( a 1 , a 2 , a 3 , a 4 , a 5 , a 6 ) = ( a 1 , a 2 , R 1 ( a 3 , a 5 , a 6 ) , a 4 , R 2 ( a 3 , a 5 , a 6 ) , R 3 ( a 3 , a 5 , a 6 )) = ( a 1 , a 2 , a ′ 3 , a 4 , a ′ 5 , a ′ 6 ) , where R ( x , y , z ) = ( R 1 ( x , y , z ) , R 2 ( x , y , z ) , R 3 ( x , y , z )) = ( x ′ , y ′ , z ′ ) . D. Talalaev Quasi-invariants of 2-knots and quantum integrable systems
Preliminaries Tetrahedral equation Results 2-knot diagram Appendices Functional equation One distinguishes a functional tetrahedral equation, satisfied by a map on some functional field, in the example below on the field of rational functions. I depict here a famous electric solution: Φ( x , y , z ) = ( x 1 , y 1 , z 1 ); xy x 1 = x + z + xyz , y 1 = x + z + xyz , yz z 1 = x + z + xyz , related to the so called star-triangle transformation, known in electric circuits Figure : Star-triangle transformation D. Talalaev Quasi-invariants of 2-knots and quantum integrable systems
Preliminaries Tetrahedral equation Results 2-knot diagram Appendices Another realization Let us consider the Euler decomposition of U ∈ SO ( 3 ) and a dual one cos φ 1 sin φ 1 0 cos φ 2 0 sin φ 2 1 0 0 U = − sin φ 1 cos φ 1 0 0 1 0 0 cos φ 3 sin φ 3 − sin φ 2 cos φ 2 − sin φ 3 cos φ 3 0 0 1 0 0 � �� � � �� � � �� � X αβ [ φ 1 ] X αγ [ φ 2 ] X βγ [ φ 3 ] U = X αβ [ φ 1 ] X αγ [ φ 2 ] X βγ [ φ 3 ] = X βγ [ φ ′ 3 ] X αγ [ φ ′ 2 ] X αβ [ φ ′ 1 ] Then the transformation from the Euler angles to the dual Euler angles sin φ ′ = sin φ 2 cos φ 1 cos φ 1 + sin φ 1 sin φ 3 2 cos φ 1 cos φ 2 3 = cos φ 2 cos φ 3 cos φ ′ cos φ ′ = , 1 cos φ ′ cos φ ′ 2 2 defines a solution of the functional tetrahedral equation. D. Talalaev Quasi-invariants of 2-knots and quantum integrable systems
Preliminaries Tetrahedral equation Results 2-knot diagram Appendices 4-cube colorings Let us consider the 4-cube and its projection to a 3-dimensional space. This is a rhombo-dodecahedron divided in two ways into four parallelepipeds, corresponding to the 3-cubes of the border of the 4-cube. Figure : Tesseract D. Talalaev Quasi-invariants of 2-knots and quantum integrable systems
Preliminaries Tetrahedral equation Results 2-knot diagram Appendices One may associate to this division a problem of coloring the 2-faces of the 4-cube by elements (called colors) of some set X in such a way that the colors of the faces in each 3-cube are related by some transformation Φ : ( a 1 , a 2 , a 3 ) → ( a ′ 1 , a ′ 2 , a ′ 3 ) . There is a special way to choose the incoming and outgoing 2-faces of each 3-cube. It appears that the compatibility condition for Φ is nothing but the tetrahedral equation. D. Talalaev Quasi-invariants of 2-knots and quantum integrable systems
Preliminaries Tetrahedral equation Results 2-knot diagram Appendices Recalling 1-knots Figure : Trefoil Figure : Reidemeister moves D. Talalaev Quasi-invariants of 2-knots and quantum integrable systems
Preliminaries Tetrahedral equation Results 2-knot diagram Appendices 2-knot Definition By a 2-knot we mean an isotopy class of embeddings S 2 ֒ → R 4 . A class of examples of non-trivial 2-knots is given by the Zeeman’s [1965] twisted-spun knot, which is a generalization of the Artin spun knot. Figure : Example D. Talalaev Quasi-invariants of 2-knots and quantum integrable systems
Preliminaries Tetrahedral equation Results 2-knot diagram Appendices Diagrams To obtain a diagram of a 2-knot one takes a generic projection p to the hyperplane P in R 4 . The generic position entails that there are singularities only of the following types: double point, triple point and the Whitney point (or branch point) Figure : Singularity types D. Talalaev Quasi-invariants of 2-knots and quantum integrable systems
Preliminaries Tetrahedral equation Results 2-knot diagram Appendices Diagrams To obtain a diagram of a 2-knot one takes a generic projection p to the hyperplane P in R 4 . The generic position entails that there are singularities only of the following types: double point, triple point and the Whitney point (or branch point) Figure : Singularity types The diagram of a 2-knot is a singular surface with arcs of double points which end in triple points and branch points. This defines a graph of singular points. The additional information consists of the order of 2-leaves in intersection lines subject to the projection direction. We always work here with oriented surfaces. D. Talalaev Quasi-invariants of 2-knots and quantum integrable systems
Preliminaries Tetrahedral equation Results 2-knot diagram Appendices Roseman moves Theorem [Roseman 1998] Two diagrams represent equivalent knotted surfaces iff one can be obtained from another by a finite series of moves from the list and an isotopy of a diagram in R 3 . D. Talalaev Quasi-invariants of 2-knots and quantum integrable systems
Preliminaries Tetrahedral equation Results 2-knot diagram Appendices Roseman moves Theorem [Roseman 1998] Two diagrams represent equivalent knotted surfaces iff one can be obtained from another by a finite series of moves from the list and an isotopy of a diagram in R 3 . There is an approach due to Carter, Saito and others (2003) which produces invariants of 2-knots by means of the so called quandle cohomology. Invariants are constructed as some partition functions on the space of states which are coloring of the 2-leaves of a diagram by elements of the quandle. D. Talalaev Quasi-invariants of 2-knots and quantum integrable systems
Preliminaries Quasi-invariants Results Regular lattices and 2d quantum integrability Appendices Summary and perspectives Preliminaries 1 Tetrahedral equation 2-knot diagram Results 2 Quasi-invariants Regular lattices and 2d quantum integrability Summary and perspectives Appendices 3 Tetrahedral complex Quandle cohomology and 2-knot invariants D. Talalaev Quasi-invariants of 2-knots and quantum integrable systems
Preliminaries Quasi-invariants Results Regular lattices and 2d quantum integrability Appendices Summary and perspectives Cocycles Electric solution: x 1 = xy / ( x + z + xyz ) , y 1 = x + z + xyz , z 1 = yz / ( x + z + xyz ) . D. Talalaev Quasi-invariants of 2-knots and quantum integrable systems
Preliminaries Quasi-invariants Results Regular lattices and 2d quantum integrability Appendices Summary and perspectives Definition For a given solution Φ of the set-theoretic tetrahedral equation on the set X and a given field k we say that a function ϕ : X × 3 → k is a 3 -cocycle of the tetrahedral complex if ϕ ( a 1 , a 2 , a 3 ) ϕ ( a ′ 1 , a 4 , a 5 ) ϕ ( a ′ 2 , a ′ 4 , a 6 ) ϕ ( a ′ 3 , a ′ 5 , a ′ 6 ) = = ϕ ( a 3 , a 5 , a 6 ) ϕ ( a 2 , a 4 , a ′ 6 ) ϕ ( a 1 , a ′ 4 , a ′ 5 ) ϕ ( a ′ 1 , a ′ 2 , a ′ 3 ) . D. Talalaev Quasi-invariants of 2-knots and quantum integrable systems
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