TL;DR
This paper introduces a new operation called doubling on biracks, transforming three-colour quandles into biquandles, which can distinguish the chirality of trefoils, advancing knot classification methods.
Contribution
It presents a novel doubling operation on biracks that enhances three-colour knot invariants to distinguish mirror images of trefoils.
Findings
Biquandle coloring distinguishes right and left handed trefoils.
The method relates to homology classes of quandle and biquandle spaces.
Provides a new tool for knot chirality detection.
Abstract
The classical trefoil is famous for having a three-colouring which distinguishes it from the unknot. The three-colouring is also notorious for not distinguishing the right handed from the left handed trefoil. However with a bit of tweaking the three colours can also be used for this task. What lies behind the method is a new operation on biracks called doubling which converts the 3-colour quandle into a biquandle. Colouring with this biquandle distinguishes the right handed from the left handed trefoil. Equivalently it defines an element of the homology of the quandle or biquandle classifying space.
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