Studies on superbridge index of knots

Abstract

In this thesis we study superbridge index of knots and examine some of its properties and relations with other knot invariants. Also we calculate superbridge index for some knots.

In Chapter 1 we introduce definitions, notions and terminology and investigate some relations between superbridge index and other knot invariants. In Chapter 2 we present two moves on knots which do not increase the superbridge numbers. In Chapter 3 we examine knots having odd superbridge index and show that there are only finitely many knots with superbridge index 3. Moreover, we show that the list $\{ 3_1 ,\ 4_1 ,\ 5_2 ,\ 6_1 ,\ 6_2 ,\ 6_3 ,\ 7_2 ,\ 7_3 ,\ 7_4 , \ 8_4 ,\ 8_7,\ 8_9 \}$ contains all knots with superbridge index 3. And we get a new upper bound for superbridge index related with bridge index. In chapter 4 we consider the connected sums of two knots and introduce some results about the superbridge index for composite knots. Also we study marginal knots and strongly marginal knots and propose some problems about those knots. In Chapter 5 we supply the best known estimates of the superbridge index for all prime knots up to nine crossings.