Kvantne vozelne invariante : delo diplomskega seminarja

Grlj, Jernej (author), Strle, Sašo (mentor)

V tem delu bomo definirali pentljaste Hopfove algebre in $rv$-vozle. S pomočjo le teh bomo vpeljali kvantne vozlene invariante in s pomočjo kvantne \(\mathfrak{sl}_2\) izračunali Jonesov polinom deteljice. Ta formulacija kvantnih vozelnih invariant sledi pristopu R. v.d. Veena [Introduction to quantum invariants of knots] in D. Bar-Natana [Universal tangle invariants and docile perturbed Gaussians].

• Language: Slovenian
• Keywords: Kvazi-trikotna Hopfova algebra, pentljasta Hopfova algebra, rv-vozel, kvantna vozelna invarianta, Jonesov polinom
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2023
• PID: 20.500.12556/RUL-148248
• UDC: 515.1
• COBISS.SI-ID: 161920771
• Publication date in RUL: 05.08.2023

Secondary language

• Language: English
• Title: Quantum knot invariants

Abstract

In this thesis, we are going to define ribbon Hopf algebras and $rv$-knots. Building on these definitions, we are going to introduce quantum knot invariants. In the end, we are going to compute the Jones polynomial of the trefoil with the help of quantum $\mathfrak{sl}_2$. This formulation of quantum invariants follows that of R. van der Veen [Introduction to quantum invariants of knots] and D. Bar-Natana [Universal tangle invariants and docile perturbed Gaussians].

• Keywords: Quasi-triangular Hopf algebra, ribbon Hopf algebra, rv-knot, quantum knot invariant, Jones polynomial