Title:

Computational aspects of knots and knot transformation

In this thesis we study the computational aspects of knots and knot trans
formations. Most of the problems of recognising knot properties (such as
planarity, unknottedness, equivalence) are known to be decidable, however
for many problems their precise time or space complexity is still unknown.
On the other hand, their complexity in terms of computational power of
devices needed to recognise the knot properties was not studied yet. In this
thesis we address this problem and provide first known bounds for some
knot problems within this context. In order to estimate and characterise
complexity of knot problems represented by Gauss words, we consider vari
ous tools and mathematical models including automata models over infinite
alphabets, standard computational models and definability in logic.
In particular we show that the planarity problem of signed and unsigned
Gauss words can be recognised by a twoway deterministic register au
tomata. Then we translate this result in terms of classical computational
models to show that these problems belong to the logspace complexity class
L, Further we consider definability questions in terms of first order logic and
its extensions and show that planarity of both signed and unsigned Gauss
words cannot be expressed by a formula of firstorder predicate logic, while
extensions of firstorder logic with deterministic transitive closure operator
allow to define planarity of both signed unsigned Gauss words. Follow
ing the same line of research we provide lower and upper bounds for the
planarity problem of Gauss paragraphs and unknottedness.
In addition we consider knot transformations in terms of string rewriting
systems and provide a refined classification of Reidemeister moves formu
lated as string rewriting rules for Gauss words. Then we analyse the reach
ability properties for each type and present some bounds on the complexity
of the paths between two knot diagrams reachable by a sequence of moves of
the same type. Further we consider a class of nonisomorphic knot diagrams
generated by type I moves from the unknot and discover that the sequence
corresponding to the number of diagrams with respect to the number of
crossings is equal to a sequence related to a class of Eulerian maps with
respect to the number of edges. We then investigate the bijective mapping
between the two classes of objects and as a result we present two algo
rithms to demonstrate the transformations from one object to the other.
It is known that unknotting a knot may lead to a significant increase in
number of crossings during the transformations. We consider the question
of designing a set of rules that would not lead to the increase in the number
of crossings during knot transformations. In particular we introduce a new
set moves in this regard which can be used to substitute one of the rules
of type II that increases the number of crossings. We show that such new
moves coupled with Reidemeister moves can unknot all known examples of
complex trivial knot diagrams without increasing number of crossings.
