Expansions of quantum group invariants

Schaveling, S. (2020)

Doctoral Thesis

In my research, we developed a method to distinguish knots. A knot is a mathematical depiction of the everyday knot that occurs in ropes and cords. The distinguishing of knots is a computationally hard problem, which is equivalent with the untangling of knots. Often, knots are distinguished by a so called knot invariant. Such an invariant is a label that can be attributed to any knot, such that the label is identical for two identical knots. The theory of knots is closely related to quantum computing and lattices of atoms. The distinguishing of knots is a problem that can be solved in an exponential number of computations, in the number of crossings of a knot. Our invariants can be computed in polynomial time. This means that the number of computations is a power of the number of crossings of a knot. Another advantage of our method is that for the knots that have been calculated, the invariant was able to distinguish as the most effective methods combined. In fact, by removing unnecessary information from the conventional knot invariants, we obtained an invariant that can be computed much faster, and is as effective as the usual invariants.

• Quantum group
• Knot invariant
• Expansion
• Colored Jones polynomial
• Polynomial time
• Perturbed exponentials
• Special linear Lie algebra
• Drinfeld double
• Hopf algebra
• Quantization

• All authors: Schaveling, S.
• Supervisor: Edixhoven, S.J.
• Co-supervisor: Veen, R.I. van der
• Committee: Stevenhagen, P.; Luijk, R.M., van; Nienhuis, B.; Bar-Natan, D.; Stokman, J.V.
• Qualification: Doctor (dr.)
• Awarding Institution: Mathematical Institute (MI), Leiden University
• Date: 2020-09-01