In the mathematical field of quantum topology, the Reshetikhin–Turaev invariants (RT-invariants) are a family of quantum invariants of framed links.
Such invariants of framed links also give rise to invariants of 3-manifolds via the Dehn surgery construction.
These invariants were discovered by Nicolai Reshetikhin and Vladimir Turaev in 1991,[1] and were meant to be a mathematical realization of Witten's proposed invariants of links and 3-manifolds using quantum field theory.
-linear ribbon category comes equipped with a diagrammatic calculus in which morphisms are represented by certain decorated framed tangle diagrams, where the initial and terminal objects are represented by the boundary components of the tangle.
In this calculus, a (decorated framed) link diagram
, being a (decorated framed) tangle without boundary, represents an endomorphism of the monoidal identity (the empty set in this calculus), or in other words, an element of
are related by a sequence of Kirby moves.
Reshetikhin and Turaev [1] used this idea to construct invariants of 3-manifolds by combining certain RT-invariants into an expression which is invariant under Kirby moves.
be a ribbon Hopf algebra over a field
are represented by framed tangle diagrams with each connected component decorated by a finite dimensional representation of
In this way, each ribbon Hopf algebra
gives rise to an invariant of framed links colored by representations of
, the corresponding RT-invariant for links and 3-manifolds gives rise to the following family of link invariants, appearing in skein theory.
denote the RT-invariant obtained by decorating each component of
denotes the Kauffman polynomial of the link
-th root of unity with odd
is obtained by doing Dehn surgery on a framed link
are the numbers of positive and negative eigenvalues for the linking matrix of
Roughly speaking, the first and second bracket ensure that
is invariant under blowing up/down (first Kirby move) and the third bracket ensures that
is invariant under handle sliding (second Kirby move).
The Witten–Reshetikhin–Turaev invariants for 3-manifolds satisfy the following properties: These three properties coincide with the properties satisfied by the 3-manifold invariants defined by Witten using Chern–Simons theory (under certain normalization)[2] Pick
Witten's asymptotic expansion conjecture suggests that for every 3-manifold
is governed by the contributions of flat connections.
[4] Conjecture: There exists constants
are the finitely many different values of the Chern–Simons functional on the space of flat
The Witten's asymptotic expansion conjecture suggests that at
, the RT-invariants grow polynomially in
, in 2018 Q. Chen and T. Yang suggested the volume conjecture for the RT-invariants, which essentially says that the RT-invariants for hyperbolic 3-manifolds grow exponentially in
and the growth rate gives the hyperbolic volume and Chern–Simons invariants for the 3-manifold.