Evaluating the Jones polynomial with tensor networks.

Area of Science:

• Knot theory
• Computational topology
• Statistical mechanics

Background:

• The Jones polynomial is a fundamental invariant in knot theory, crucial for distinguishing knots.
• Evaluating the Jones polynomial for complex knots is computationally challenging with existing methods.
• The problem can be mapped to calculating partition functions in statistical mechanics models.

Purpose of the Study:

• To introduce novel tensor network contraction algorithms for computing the Jones polynomial.
• To demonstrate the efficiency and applicability of these algorithms for arbitrary knots.
• To establish tensor network methods as a practical tool in knot theory research.

Main Methods:

• Formulating the Jones polynomial evaluation as a partition function of a q-state anisotropic Potts model.
• Casting the Potts model partition function into a tensor network form.
• Employing fast tensor network contraction protocols for exact evaluation.

Main Results:

• The Jones polynomial evaluation is shown to be equivalent to a specific tensor network contraction.
• Numerical simulations on random knots demonstrate subexponential time scaling for evaluation.
• The method successfully computes the Jones polynomial for knots too complex for other techniques.

Conclusions:

• Tensor network contraction algorithms provide an efficient method for Jones polynomial evaluation.
• This approach significantly expands the scope of computationally tractable knot structures.
• Tensor network methods are established as a practical and powerful tool for knot theory.