Kvanteinvariants

Kvanteinvariants, also known as quantum invariants, are quantities assigned to knots, links, and 3-manifolds that remain unchanged under topological or smooth deformations. They arise from ideas in quantum physics and representation theory and provide powerful tools in low-dimensional topology.

Theoretical foundation: They come from quantum groups U_q(g) and from Chern-Simons topological quantum field theory. Using representations of quantum groups one can construct link invariants through R-matrices and skein relations; for manifolds, the Reshetikhin-Turaev construction yields invariants of links in 3-manifolds and of the manifolds themselves. When q is specialized to roots of unity, the resulting invariants underpin 3-dimensional TQFTs (topological quantum field theories) and modular tensor categories; the Turaev-Viro invariants give a complementary state-sum approach.

Examples: The Jones polynomial is a quantum invariant derived from U_q(sl2). The HOMFLY-PT polynomial extends Jones as another quantum invariant. More general quantum invariants include Witten-Reshetikhin-Turaev invariants and Kauffman bracket relations. These invariants distinguish knots and links and provide information about their properties such as sliceness and genus in some contexts.

Applications and significance: Quantum invariants have deep connections to low-dimensional topology, representation theory, algebraic geometry, and quantum computing. They illustrate how quantum theory can be used to study classical geometric structures and lead to computational approaches via tensor categories and state-sum models.