No description available.

No description available.

This thesis uses Kauffman skein theory to give several new results. We show a correspondence between Kauffman and Homily satellite invariants with coefficients modulo 2, when we take certain patterns from the respective skeins of the annulus. Using stacked tangles we construct a polynomial time algorithm for calculating the Kauffman polynomial of links, and then extend the theory to give a new polynomial time algorithm for calculating the Homfly polynomial. We show that the Kauffman polynomials of genus 2 mutants can differ, and improve on existing examples showing the non-invariance of the Homfly polynomial under genus 2 mutation. By expressing twists as single crossings and smoothings in the Kauffman skein we develop an algorithm for calculating the Kauffman polynomial of pretzel links. Finally we consider the result of some calculations in the Kauffman skein of the annulus.

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 two-way deterministic register au- tomata. Then we translate this result in terms of classical computational models to show that these problems belong to the log-space 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 first-order predicate logic, while extensions of first-order 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 non-isomorphic 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.

No description available.

We study Seifert surfaces for links, and in particular the Kakimizu complex MS(L) of a link L, which is a simplicial complex that records the structure of the set of taut Seifert surfaces for L. First we study a connection between the reduced Alexander polynomial of a link and the uniqueness of taut Seifert surfaces. Specifically, we reprove and extend a particular case of a result of Juhasz, using very different methods, showing that if a non-split homogeneous link has a reduced Alexander polynomial whose constant term has modulus at most 3 then the link has a unique incompressible Seifert surface. More generally we see that this constant term controls the structure of any non-split homogeneous link. Next we give a complete proof of results stated by Hirasawa and Sakuma, describing explicitly the Kakimizu complex of any non-split, prime, special alternating link. We then calculate the form of the Kakimizu complex of a connected sum of two non-fibred links in terms of the Kakimizu complex of each of the two links. This has previously been done by Kakimizu when one of the two links is fibred. Finally, we address the question of when the Kakimizu complex is locally infinite. We show that if all the taut Seifert surfaces are connected then MS(L) can only be locally infinite when L is a satellite of a torus knot, a cable knot or a connected sum. Additionally we give examples of knots that exhibit this behaviour. We finish by showing that this picture is not complete when disconnected taut Seifert surfaces exist.