Horizon Academic is proud to announce our new mathematics course, Theoretical Mathematics: Studying Knots, Links, Invariants to Prove Causality. Taught by tenured Dartmouth professor, Vladimir Chernov, this course allows high school students to research advanced mathematical topics and to further develop their cross-functional analytical skills.

Knot Theory is the mathematical branch of topology with many applications, such as analyzing DNA structures. In this theoretical math course, we study knot theory with an eye for one of the most foundational uses of all: understanding causality and the relationship between events. Identifying cause and effect is foundational: to understanding the physics that govern our universe to functioning in daily life. Two points or events in a spacetime are causally related if one can get from one of them to the other without exceeding the speed of light. Mathematicians and physicists have related causality in spacetimes to the study of knots and links, and knot theorists often study simplified, flatted models of knots we encounter in everyday life, such as shoelaces tied together, to develop models for how we might understand causality and complex objects. We will analyze models of knots and links in 2+1-dimensional spacetimes and apply computable link invariants to study what invariants can plausibly enable us to detect causality between two points or events, drawing on the work in particular of Samantha Allen and Jacob Swenberg. Students will be free to select their own research topic relevant to knot theory, but Prof. Chernov is able to recommend particular racks or quandles that students can analyze. Student projects will examine which (of the many available) quandle invariants can be combined to the Alexander-Conway polynomial in order to plausibly detect causality in the toy models of the 2+1 dimensional spacetimes.

Detailed Course Description

Student projects will build on major developments in knot theory, culminating in their independent research topics. In order to reach this, students will examine the following theoretical developments:

- Robert Low (a student of this year Nobel Prize Winner Sir Roger Penrose) posed a conjecture relating causality in simplified, flatted models, called toy models, of (2+1)-dimensional spacetimes to the study of knots and links, essentially circular shoelaces tied together in different configurations. The Low conjecture was expanded on by Jose Natario and Paul Tod in the Legendrian Low conjecture to examine real world (3+1)-dimensional spacetimes and led to the question communicated by Penrose on the Vladimir Arnold Problem List. These conjectures and the questions were solved in the works of Stefan Nemirovski and Vladimir Chernov.
- In order to be able to apply these results to the real life problems, one needs to have computable invariants of links that completely determine causality. The work of Gage Martin, Ina Petkova and Vladimir Chernov show that the very powerful but computable Heegaard-Floer and Khovanov Homology Theories do solve this problem for the toy models of the (2+1)-dimensional spacetimes.
- The very recent work of Samantha Allen and Jacob Swenberg studied the question of whether the Alexander-Conway polynomial and the Jones polynomial are enough for this purpose. These polynomial invariants are obtained from the above homology theories by omitting much information, and the results of Allen and Swenberg suggest that the Jones polynomial is enough to detect causality, but the Alexander-Conway polynomial is likely not enough.
- Quandles and Racks are the classical and somewhat technical, but computable, link invariants that generalize the tri-coloring invariant. In this course, we will discuss all the theories mentioned above, and students will develop projects exploring which of the many Quandle invariants should be added to the Alexander-Conway polynomial so that it becomes plausible that they together completely detect causality in toy models of (2+1)-dimensional spacetimes.