To attack certain problems in 4-dimensional knot theory the author draws on a variety of techniques, focusing on knots in S^T4, whose fundamental groups contain abelian normal subgroups. Their class contains the most geometrically appealing and best understood examples. Moreover, it is possible to apply work in algebraic methods to these problems. Work in four-dimensional topology is applied in later chapters to the problem of classifying 2-knots.

The study of smooth embeddings of 3-manifolds in 4-space has been hampered by difficulties with the simplest case, that of homology spheres. This book presents some advantages of working with locally flat embeddings. The first two chapters outline the tools used and give general results on embeddings of 3-manifolds in S4. The next two chapters consider which Seifert manifolds may embed, with criteria in terms of Seifert data. After summarizing results on those Seifert manifolds that embed smoothly, the following chapters determine which 3-manifolds with virtually solvable fundamental groups embed. The final three chapters study the complementary regions. When these have 'good' fundamental groups, topological surgery may be used to find homeomorphisms. Figures throughout help illustrate links representing embeddings and open questions are further discussed in the appendices, making this a valuable resource for graduate students and research workers in geometric topology.

This book serves as a reference on links and on the invariants derived via algebraic topology from covering spaces of link exteriors. It emphasizes the features of the multicomponent case not normally considered by knot-theorists, such as longitudes, the homological complexity of many-variable Laurent polynomial rings, the fact that links are not usually boundary links, free coverings of homology boundary links, the lower central series as a source of invariants, nilpotent completion and algebraic closure of the link group, and disc links. Invariants of the types considered here play an essential role in many applications of knot theory to other areas of topology.This second edition introduces two new chapters — twisted polynomial invariants and singularities of plane curves. Each replaces brief sketches in the first edition. Chapter 2 has been reorganized, and new material has been added to four other chapters.

To attack certain problems in 4-dimensional knot theory the author draws on a variety of techniques, focusing on knots in S^T4, whose fundamental groups contain abelian normal subgroups. Their class contains the most geometrically appealing and best understood examples. Moreover, it is possible to apply work in algebraic methods to these problems. Work in four-dimensional topology is applied in later chapters to the problem of classifying 2-knots.

The study of smooth embeddings of 3-manifolds in 4-space has been hampered by difficulties with the simplest case, that of homology spheres. This book presents some advantages of working with locally flat embeddings. The first two chapters outline the tools used and give general results on embeddings of 3-manifolds in S4. The next two chapters consider which Seifert manifolds may embed, with criteria in terms of Seifert data. After summarizing results on those Seifert manifolds that embed smoothly, the following chapters determine which 3-manifolds with virtually solvable fundamental groups embed. The final three chapters study the complementary regions. When these have 'good' fundamental groups, topological surgery may be used to find homeomorphisms. Figures throughout help illustrate links representing embeddings and open questions are further discussed in the appendices, making this a valuable resource for graduate students and research workers in geometric topology.

This book serves as a reference on links and on the invariants derived via algebraic topology from covering spaces of link exteriors. It emphasizes the features of the multicomponent case not normally considered by knot-theorists, such as longitudes, the homological complexity of many-variable Laurent polynomial rings, the fact that links are not usually boundary links, free coverings of homology boundary links, the lower central series as a source of invariants, nilpotent completion and algebraic closure of the link group, and disc links. Invariants of the types considered here play an essential role in many applications of knot theory to other areas of topology.This second edition introduces two new chapters — twisted polynomial invariants and singularities of plane curves. Each replaces brief sketches in the first edition. Chapter 2 has been reorganized, and new material has been added to four other chapters.