An Enhanced Decomposition Theorem for Knots with Symmetry Information

TL;DR

This paper introduces an advanced prime decomposition theorem for knots that incorporates symmetry information, enabling detailed classification and construction of composite knots from prime factors, with a comprehensive table up to 12 crossings.

Contribution

It provides an enhanced algebraic decomposition theorem for knots that includes symmetry considerations and offers a detailed table of composite knots up to 12 crossings.

Findings

Decomposition theorem accounts for mirror and orientation symmetries.

Constructs a table of composite knots up to 12 crossings.

Relies on conjectural additivity of crossing number for completeness.

Abstract

We present an enhanced prime decomposition theorem for knots that gives the isotopy classes of composite knots that can be constructed from a given list of prime factors (allowing for the mirroring and orientation reversing for each factor). Underlying the theorem is an algebraic construction that also allows for the computation of the intrinsic symmetries (invertibility, chirality, etc.) of a composite knot from those of the prime factors. We then use this construction to give a table of composite knots through 12 crossings that can be constructed from prime factors through 9 crossings. This is more difficult than it might sound because we must take knot symmetries into account when generating the table (the square knot and the granny knot are different, though both are connect sums of two trefoils). The completeness of this table depends on the conjectural additivity of crossing number under the connected sum.