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We introduce textit{Kaestner brackets}, a generalization of biquandle brackets to the case of parity biquandles. This infinite set of quantum enhancements of the biquandle counting invariant for oriented virtual knots and links includes the classical quantum invariants, the quandle and biquandle $2$-cocycle invariants and the classical biquandle brackets as special cases, coinciding with them for oriented classical knots and links but defining generally stronger invariants for oriented virtual knots and links. We provide examples to illustrate the computation of the new invariant and to show that it is stronger than the classical biquandle bracket invariant for virtual knots.
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Biquandle brackets are a type of quantum enhancement of the biquandle counting invariant for oriented knots and links, defined by a set of skein relations with coefficients which are functions of biquandle colors at a crossing. In this paper we use biquandle brackets to enhance the biquandle counting matrix invariant defined by the first two authors in arXiv:1803.11308. We provide examples to illustrate the method of calcuation and to show that the new invariants are stronger than the previous ones.
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