Topologically non-trivial field excitations, including the ones carrying linked and knotted
structures, play important roles in many physical systems ranging from classical fluids to
liquid crystals, plasmas, electromagnetism, and to fundamental quantum fields. These excitations
can appear spontaneously during symmetry breaking phase transitions. For example, in cosmological
theories cosmic strings may have formed knotted configurations influencing the Early Universe
development while in liquid crystals transient knotted defect lines are observed during
isotropic-to-nematic transition. In the latter case knotted topological defects appear
spontaneously and are transient non-equilibrium field configurations, which eventually relax
to equilibrium defect-free states.

In collaboration with the experimental group
of Prof. Smalyukh at the University of Colorado Boulder we explore how topologically non-trivial
spatial confinements can be used in order to achieve a robust control of the appearance and stability
of topological field excitations. In order to achieve this goal, we use a nematic liquid crystal as a
model system and confine it by topologically non-trivial surfaces with systematically varied genus.
This allows to generate topological defects of the desired total hedgehog charge which obeys
predictions of the Gauss-Bonnet and Poincare-Hopf index theorems. Complementary theoretical calculations
and experimental observations reveal non-trivial structures and transformations of defect lines as a
function of the surface topology and material and geometric parameters, establishing a robust means of
controlling solitonic, knotted, linked and other field excitations. Since few theoretical predictions
of topological field configurations can be tested experimentally due to lack of experimentally accessible
systems and techniques, our model system may become a testbed for probing a potentially scale-invariant
interplay of topologies of confining surfaces, fields, and defects. Similar to probing the cosmological
Kibble mechanism using liquid crystal
phase transitions, it may enable new cosmology and particle physics relevant experiments.