PHYSICAL AND GEOMETRIC KNOT THEORY: THE TOPOLOGY-GEOMETRY INTERPLAY AND APPLICATIONS

This book develops the theory of physical and geometric knot theory, presenting a systematic study of knots and links as geometric objects subject to physical constraints. A central theme of the book is the quantitative study of thick knots and links, with emphasis placed on ropelength and related geometric invariants, and substantial attention devoted to establishing sharp upper and lower bounds. The authors develop methods that connect geometric quantities to classical knot invariants, including braid index, bridge index, and knot polynomials, thereby revealing deep interactions between topology and geometry.Beyond foundational theory, the book explores modern developments in random knot models and the topology of DNA and polymers, illustrating how geometric knot theory provides insight into molecular entanglement, the statistical behavior of long chains, and physical constraints in real systems.Bridging the gap between classical, abstract knot theory and the tangible realities of how knots behave, form, and function in nature and applications, the exposition is grounded in rigorous mathematics while consistently motivated by phenomena arising in biology, physics, and materials science.Intended for researchers and graduate students in mathematics and related disciplines, this volume offers both a comprehensive treatment of geometric knot theory and a framework for future advances at the interface of topology, geometry, and the natural sciences.