Mathematicians Mark Brittenham and Sarah Hermiller have refuted the additivity conjecture in knot theory, showing that the unknotting number of a connect sum of two knots is not always the sum of their individual unknotting numbers. Their discovery, based on a database of unknotting numbers and computational methods, challenges assumptions about the structure of knots. This finding highlights the unpredictability of crossing changes and opens new research directions in knot theory. It also underscores the role of computational power in modern mathematical research. The result is worth reading because it reveals the complexity of knot invariants and the need for deeper exploration of non-additive behavior in knots. Readers will learn how computational methods can uncover long-standing assumptions in mathematics.
Key facts
- Mathematicians Mark Brittenham and Sarah Hermiller found a counterexample to the additivity conjecture in knot theory.
- The connect sum of two (2,7) torus knots, each with an unknotting number of 3, can be undone in just 5 crossing changes, not 6.
- The discovery used a database of unknotting numbers built over a decade using supercomputers and distributed computing.
- The result shows that the unknotting number is not always additive, challenging assumptions about the structure of knot invariants.
- The research highlights the growing role of computational methods in solving complex mathematical problems.
