# Abstract/Details

## Some results in formal knot theory

1994 1994

### Abstract (summary)

In this thesis we study pseudo link diagrams of two kinds (both generalizing link diagrams) in formal knot theory.

In Chapter 0 we introduce some necessary terms and conventions.

In Chapter 1 we define (oriented) spatial ${\bf R}\sp2$-diagrams which generalize (oriented) link diagrams in the way that strings besides loops are allowed in such a (oriented) diagram, introduce their certain transformations called (oriented) spatial ${\bf R}\sp2$-moves generalizing (oriented) Reidemeister moves and then study the equivalence relation on (oriented) spatial ${\bf R}\sp2$-diagrams called slide-equivalence of (oriented) spatial ${\bf R}\sp2$-diagrams generated by part of (oriented) spatial ${\bf R}\sp2$-moves. Their study results in the complete classification of slide-equivalence classes of (oriented) spatial ${\bf R}\sp2$-diagrams. In particular, link diagrams belong to two slide-equivalence classes and oriented link diagrams belong to infinite slide-equivalence classes. In fact, we obtain similar results for (oriented) spatial $M\sp2$-diagrams where $M\sp2$ is an arbitrary surface.

In Chapter 2 we first define (oriented) link S-shadows which generalize (oriented) link diagrams in the way that each crossing of such a (oriented) diagram has a value in an arbitrary value set S instead of an over/under structure, introduce their certain transformations generalizing (oriented) Reidemeister moves, and study isotopy and regular isotopy of (oriented) S-shadows defined similarly. Their study results in that there is a regular isotopy invariant of link S-shadows generalizing the Kauffman bracket and it can be normalized into an isotopy invariant of oriented link S-shadows generalizing the normalized Kauffman bracket. We then define braid S-shadows generalizing braid diagrams in a similar way, introduce their certain transformations generalizing braid moves, and study isotopy of braid S-shadows defined similarly. Their study results in that isotopy classes of n-braid S-shadows form a group which generalizes the n-braid group $B\sb{n}$ and has a monoid representation generalizing the usual monoid representation of $B\sb{n}$ into the n-Temperly-Lieb algebra.