In their paper entitled “Quantum Enhancements and Biquandle Brackets”, Nelson, Orrison, and Rivera introduced biquandle brackets, which are customized skein invariants for biquandle-colored links. We prove herein that if a biquandle bracket [Formula: see text] is the pointwise product of the pair of functions [Formula: see text] with a function [Formula: see text], then [Formula: see text] is also a biquandle bracket if and only if [Formula: see text] is a biquandle 2-cocycle (up to a constant multiple). As an application, we show that a new invariant introduced by Yang factors in this way, which allows us to show that the new invariant is in fact equivalent to the Jones polynomial on knots. Additionally, we provide a few new results about the structure of biquandle brackets and their relationship with biquandle 2-cocycles.

Spectral clustering is discussed from many perspectives, by extending it to rectangular arrays and discrepancy minimization too. Near optimal clusters are obtained with singular value decomposition and with the weighted k-means algorithm. In case of rectangular arrays, this means enhancing the method of correspondence analysis with clustering, and in case of edge-weighted graphs, a normalized Laplacian based clustering. In the latter case it is proved that a spectral gap between the (k − 1)th and kth smallest positive eigenvalues of the normalized Laplacian matrix gives rise to a sudden decrease of the inner cluster variances when the number of clusters of the vertex representatives is 2 k−1 , but only the first k − 1 eigenvectors, constituting the so-called Fiedler-carpet, are used in the representation. Application to directed migration graphs is also discussed.

We discuss spectral clustering from a variety of perspectives that include extending techniques to rectangular arrays, considering the problem of discrepancy minimization, and applying the methods to directed graphs. Near-optimal clusters can be obtained by singular value decomposition together with the weighted k k -means algorithm. In the case of rectangular arrays, this means enhancing the method of correspondence analysis with clustering, while in the case of edge-weighted graphs, a normalized Laplacian-based clustering. In the latter case, it is proved that a spectral gap between the ( k − 1 ) \left(k-1) st and k k th smallest positive eigenvalues of the normalized Laplacian matrix gives rise to a sudden decrease of the inner cluster variances when the number of clusters of the vertex representatives is 2 k − 1 {2}^{k-1} , but only the first k − 1 k-1 eigenvectors are used in the representation. The ensemble of these eigenvectors constitute the so-called Fiedler-carpet.

The Biquandle Bracket is a generalization of the Jones Polynomial. In this paper, we outline a Khovanov Homology-style construction which generalizes Khovanov Homology and attempts to categorify the Biquandle Bracket. The Biquandle Bracket is not always recoverable from our construction, so this is not a true categorification. However, this deficiency leads to a new invariant: a canonical biquandle 2-cocycle associated to a biquandle bracket.