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Title of the thesis: Algorithms and computations in Khovanov homology

Thesis defender: Tuomas Kelomäki 
Opponent: Professor Vjacheslav Krushkal, University of Virginia, USA
Custos: Associate Professor Kalle Kytölä, Aalto University School of Science

Knot theory is a branch of mathematics that studies how pieces of strings can be intertwined with themselves and with each other. Mathematically, this is formulated as the study of loops smoothly embedded in a three dimensional space. It has been known for a hundred years that these smooth embeddings, knots, can be represented as simple combinatorial diagrams and hence one can study them using combinatorial methods. 

Early combinatorial tools of studying knots took values in polynomials. At the beginning of millennium, these polynomial invariants were upgraded into multigraded homology theories. The upgrades, aptly named categorifications, generated stronger invariants which have also allowed knot theorists to poke at 4-dimensional topology by creating obstructions for diffeomorphisms. The downside of the new exciting invariants is that they are harder to compute both algorithmically and theoretically. 

This dissertation, which lies at the intersection of topology, algebra, and combinatorics, examines the computational properties of Khovanov homology, a categorification of the Jones polynomial. Discrete Morse theory, a tool from combinatorial topology which is new in the context of knot homologies, is used to examine and produce new results on Khovanov homology of several infinite families of links. In addition to theoretical calculations, new results on the computational complexity of Khovanov homology are proven in the dissertation.

Keywords: Khovanov homology, low-dimensional topology, combinatorics, homological algebra

Thesis available for public display 7 days prior to the defence at . 

Contact Information:
tuomas.kelomaki@aalto.fi