# Invitation to Topological Robotics

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*Michael Farber*

A publication of the European Mathematical Society

This book discusses several selected topics of a new emerging area of
research on the interface between topology and engineering. The
first main topic is topology of configuration spaces of
mechanical linkages. These manifolds arise in various fields of
mathematics and in other sciences, e.g., engineering, statistics,
molecular biology. To compute Betti numbers of these configuration
spaces the author applies a new technique of Morse theory in the presence of an
involution. A significant result of topology of linkages presented in
this book is a solution of a conjecture of Kevin Walker which states that
the relative sizes of bars of a linkage are determined, up to certain
equivalence, by the cohomology algebra of the linkage configuration
space.

This book also describes a new probabilistic approach to topology of
linkages which treats the bar lengths as random variables and studies
mathematical expectations of Betti numbers. The second main topic is
topology of configuration spaces associated to polyhedra. The author
gives an account of a beautiful work of S. R. Gal, suggesting an
explicit formula for the generating function encoding Euler
characteristics of these spaces. Next the author studies the knot
theory of a robot arm, focusing on a recent important result of
R. Connelly, E. Demain, and G. Rote. Finally, he investigates
topological problems arising in the theory of robot motion planning
algorithms and studies the homotopy invariant TC(X)
measuring navigational complexity of configuration spaces.

This book is intended as an appetizer and will introduce the reader to
many fascinating topological problems motivated by engineering.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

#### Readership

Graduate students and research mathematicians interested in topology.