The Speakers

Title: Generalized volume conjecture for 3-manifolds, BPS spectra, and black swans

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Title: Entanglement entropy and colored Jones polynomial

Abstract: The colored Jones polynomial assigns a quantum state to an n-component link. This state lives in the n-fold tensor product of the Hilbert space of Chern-Simons theory on a torus. We study the interplay between the quantum entanglement structure of this state and the topological entanglement structure of the link. For example, the unlink is quantumly unentangled, while the Hopf link is maximally entangled. In Abelian Chern-Simons theory, it is possible to give a general formula for the entanglement entropy of any sublink of a bigger link in terms of the Gauss linking matrix. In non-Abelian Chern-Simons theory, we show that all torus links correspond to generalized GHZ states, and discuss the entanglement structure of hyperbolic links in terms of entanglement negativity. We end with some interesting open directions on connections to SL(2,C) Chern-Simons theory and the volumes of hyperbolic link complements. 

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Title: Can we efficiently compute Chern-Simons invariants?

Abstract: In this talk, I will explore the connection between knot theory and Chern-Simons quantum field theory. I will review the method for obtaining polynomial invariants for knots within the Chern-Simons framework. Additionally, I will discuss various techniques for computing these invariants and compare direct and alternative approaches. Finally, I will address the challenges of achieving computational efficiency and examine how optimized algorithms can enhance our understanding of Chern-Simons invariants.

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Title: Learning knot invariance

Abstract:  Knots are embedded circles in a R3 and are considered equivalent if related by ambient isotopy. We propose to use techniques from generative AI and contrastive learning to automate the process of learning knot invariance. We set up a neural network with a contrastive loss that clusters different representations from the same knot equivalence class in the embedding dimension. We also use transformers to map different representations from the same knot equivalence class to a single (arbitrary) representative of their class. We explain how to use the generative model to study the Jones unknotting conjecture and how we examine which invariants are learned by the trained model.

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Title: Big data approach to knot invariants

Abstract:   In 1987, Ernst and Summers demonstrated that the number of knots increases exponentially with the number of crossings, effectively categorizing knots as big data. Consequently, big data techniques offer a novel approach to analyzing knots and their invariants. This talk will focus on employing tools from applied algebraic topology and AI to explore the space of knots and their invariants.

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Title:  A Polynomial-Time Algorithm to Compute the Isotopy Class of Positively-Reducible Knots

Abstract: In this talk, we provide a complete invariant of knot diagrams in terms of sequences that are equivalent to oriented Gauss codes. Then introduce relations, K-moves, that correspond to Reidemeister moves which are computed as discrete operations on our sequences. This recasts the class of equivalent knot diagrams to a class of sequences modulo K-moves. Furthermore, we display algorithms that can exactly determine whether two sequences are equivalent with the condition of being Positively-Reducible, presenting a method to compute the class of a class of Knot Diagrams in polynomial-time. We then prove that the algorithm can be computed in the order of $O(n^4)$, and finally compare our algorithm to existing algebraic methods and polynomial invariants displaying its significant improvement in computational complexity.

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Title:  Colored Jones Polynomials and the Volume Conjecture

Abstract:  We calculate and analyse a large dataset of adjoint Jones polynomials of hyperbolic knots up to 15 crossings. I will first review our method to calculate colored Jones polynomials. I will then present results from numerical explorations of our dataset of adjoint Jones polynomials. Using neural networks, we find that data about adjoint Jones polynomials is sufficient to predict the volumes of knots with an accuracy of 99.6%, a significant improvement upon the accuracy of 98.3% of predictions using fundamental Jones polynomials. I will end with some comments about our results and their potential implications for the volume conjecture.

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Title:  Representations of knots for applications in machine learning

Abstract:  Knots form an infinite and complex data set, with topological invariants that are often intertwined in ways not yet fully understood. Many foundational challenges in knot theory and low-dimensional topology can be recast as problems in reinforcement learning and generative machine learning. A key decision in approaching knot theory through an ML lens is determining how to represent knots in a machine-readable format, which can be thought of as selecting a suitable prior distribution over the space of all knots. In this talk, I will explore the challenges of representing knots for ML applications and showcase recent examples where machine learning has been successfully applied to problems in low-dimensional topology.

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Title: Searching for Ribbons with Machine Learning

Abstract: We apply Bayesian optimization and reinforcement learning to a problem in topology: the question of when a knot bounds a ribbon disk. This question is relevant in an approach to disproving the four-dimensional smooth Poincaré conjecture; using our programs, we rule out many potential counterexamples to the conjecture. We also show that the programs are successful in detecting many ribbon knots in the range of up to 70 crossings.

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