nLab mutant knot

Idea

In knot theory, the known problem of unwinding? knots gives rise to the factorization of knots generated by composing prime knots and their mutants. The computational process of computing knot invariants, namely the polynomials describing these invariants, has been postulated as a basis for “attacks” in quantum cryptography. Classes of knots called mutants would hinder these attacks and remain undetected by a quantum computer, at their present maturity. The mutants of a given knot are indistinguishable since they all have the same Jones’ type invariants. Mutants also appear in the field-theoretic framework manifest in the expectation values of Wilson loop operators.

Definition

Beginning with the tangle representation of an oriented knot? $K$, one can obtain a mutant $K'$ by replacing one of the tangles with a rotated version of the former tangle. It is important to note that the process of mutation can be executed on various subsets of the same knot diagram containing at least one crossing?.

References

• Marzuoli and Palumbo, Post Quantum Cryptography from Mutant Prime Knots, 2011, Int. J. Geom. Methods Mod. Phys. (arXiv: 1010.2055/math-ph)