Rishabh Karthikeyan Hariharan - Academia.edu (original) (raw)

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Papers by Rishabh Karthikeyan Hariharan

Knot theory is a fascinating branch of mathematics that delves into the study of loops in three-d... more Knot theory is a fascinating branch of mathematics that delves into the study of loops in three-dimensional space. By examining how these loops can be twisted, entangled, and deformed without cutting or breaking, knot theory provides profound insights into the structure of our universe, ranging from the microscopic strands of DNA to the complex topology of quantum fields. With its roots in 19th-century mathematics and its influence extending to modern physics, biology, and computer science, knot theory bridges pure mathematical abstraction and practical applications. This paper offers a comprehensive exploration of knot theory, starting with its foundational concepts and definitions. It examines the tools mathematicians use to classify knots, including powerful invariants like the Jones polynomial and knot groups. The paper also highlights advanced topics, such as the relationship between braids and knots and the role of knot theory in higher dimensions. Finally, it sheds light on the practical relevance of knot theory, showcasing its applications in understanding DNA replication, quantum computing, and fluid dynamics. By weaving together intuition, rigorous mathematics, and real-world applications, this paper aims to provide both an introduction to knot theory and a glimpse into the open problems that continue to challenge and inspire mathematicians. Whether you're a mathematician, a scientist, or simply a curious mind, this journey through the tangled world of knots promises to be as intriguing as the knots themselves.

Knot theory is a fascinating branch of mathematics that delves into the study of loops in three-d... more Knot theory is a fascinating branch of mathematics that delves into the study of loops in three-dimensional space. By examining how these loops can be twisted, entangled, and deformed without cutting or breaking, knot theory provides profound insights into the structure of our universe, ranging from the microscopic strands of DNA to the complex topology of quantum fields. With its roots in 19th-century mathematics and its influence extending to modern physics, biology, and computer science, knot theory bridges pure mathematical abstraction and practical applications. This paper offers a comprehensive exploration of knot theory, starting with its foundational concepts and definitions. It examines the tools mathematicians use to classify knots, including powerful invariants like the Jones polynomial and knot groups. The paper also highlights advanced topics, such as the relationship between braids and knots and the role of knot theory in higher dimensions. Finally, it sheds light on the practical relevance of knot theory, showcasing its applications in understanding DNA replication, quantum computing, and fluid dynamics. By weaving together intuition, rigorous mathematics, and real-world applications, this paper aims to provide both an introduction to knot theory and a glimpse into the open problems that continue to challenge and inspire mathematicians. Whether you're a mathematician, a scientist, or simply a curious mind, this journey through the tangled world of knots promises to be as intriguing as the knots themselves.