by Anurakti Gupta
A Thesis
submitted to
Indian Institute of Science Education and Research Pune
in partial fulfillment of the requirements for the
BS-MS Dual Degree Programme
Abstract
In a recent paper Champanerkar, Kofman and Purcell defined
right-angled volume for prime alternating links as a sum of volumes of
an associated collection of hyperbolic right-angled ideal polyhedra
which is an invariant of the alternating link. Around the same time
Felsner and Rote gave a graph theoretic algorithm to obtain
right-angled circle patterns associated to planar graphs. In this
thesis, we extend the Felsner-Rote algorithm to alternating knot and
link diagrams by developing graph theoretic analogs of the two moves
used to compute right-angled volumes, namely rational reduction and
decomposition along prismatic 4-circuits. Using this technique we
compute right-angled volume for knots in the alternating knot census
up to 17 crossings, and links in the alternating link census up to 14
crossings. In addition, using our methods we extend computations of
right-angled volume of weaving knots and links, verify their
conjecture on the existence of right-angled knots for alternating
knots up to 17 crossings, give a new method to generate volumes of
right-angled polyhedra recreating volumes computed by Vesnin and
Egorov, and explore volumes of fully augmented link complements.
Data of right-angled volume for alternating links computed using our algorithms. (in html format)
Census knots and links
Note that alternating knots with crossing number less than or equal to 7, and alternating links with crossing number less than or equal to 5 are all rational and hence have zero right-angled volume.
Weaving links
A weaving link W(m,n) is an alternating link diagram which has the
same projection as the torus link T(m,n). Weaving link diagrams are
rationally reduced and have no prismatic 4-circuits. Here is the data
of right-angled volume for weaving links W(m,n) we computed using our
algorithms.
Comparison with volumes of ideal right-angled polyhedra computed by Vesnin-Egorov
LINK TO BE POSTED