Knot Theory: From Fox 3-Colorings of Links to Yang–Baxter Homology and Khovanov Homology

Przytycki, Józef H.

doi:10.1007/978-3-030-16031-9_5

Cited by 2 publications

(2 citation statements)

References 29 publications

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“…, k} is always denoted by M (i,j,...,k) . Particularly, M (1,2,3,4) = M † , called the Moore-Penrose inverse of M, which always exists and is unique. Furthermore, there is a unique inverse M (2,5,1 k ) of a square matrix M called the Drazin inverse, and M D is its label.…”

Section: Preliminaries and Motivationmentioning

confidence: 99%

“…Equation (1) possesses a similar format to the famous Yang-Baxter equation, first introduced by Yang [1] in 1967 and then by Baxter [2] independently in 1972, in the field of statistical mechanics. The classic Yang-Baxter equation has been a hot research area in science and engineering applications, closely related to various mathematical subjects, such as knot theory [3], braid groups [4], statistical mechanics [5], and quantum research [6]. So, it is necessary to find partial or general solutions of (1) from the viewpoint of matrix theory.…”

Section: Introduction and Literature Reviewmentioning

confidence: 99%

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Commuting Outer Inverse-Based Solutions to the Yang–Baxter-like Matrix Equation

et al. 2022

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This paper investigates new solution sets for the Yang–Baxter-like (YB-like) matrix equation involving constant entries or rational functional entries over complex numbers. Towards this aim, first, we introduce and characterize an essential class of generalized outer inverses (termed as {2,5}-inverses) of a matrix M, which commute with M. Towards this aim, first, we introduce and characterize an essential class of generalized outer inverses (termed as {2,5}-inverses) of a matrix, which commute with it. This class of {2,5}-inverses is defined based on resolving appropriate matrix equations and inner inverses. In general, solutions to such matrix equations represent optimization problems and require the minimization of corresponding matrix norms. We decided to analytically extend the obtained results to the derivation of explicit formulae for solving the YB-like matrix equation. Furthermore, algorithms for computing the solutions are developed corresponding to the suggested methods in some computer algebra systems. The main features of the proposed approach are highlighted and illustrated by numerical experiments.

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Section: Preliminaries and Motivationmentioning

confidence: 99%

Section: Introduction and Literature Reviewmentioning

confidence: 99%

Commuting Outer Inverse-Based Solutions to the Yang–Baxter-like Matrix Equation

et al. 2022

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Zeroing Neural Network Approaches Based on Direct and Indirect Methods for Solving the Yang–Baxter-like Matrix Equation

et al. 2022

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This research introduces three novel zeroing neural network (ZNN) models for addressing the time-varying Yang–Baxter-like matrix equation (TV-YBLME) with arbitrary (regular or singular) real time-varying (TV) input matrices in continuous time. One ZNN dynamic utilizes error matrices directly arising from the equation involved in the TV-YBLME. Moreover, two ZNN models are proposed using basic properties of the YBLME, such as the splitting of the YBLME and sufficient conditions for a matrix to solve the YBLME. The Tikhonov regularization principle enables addressing the TV-YBLME with an arbitrary input real TV matrix. Numerical experiments, including nonsingular and singular TV input matrices, show that the suggested models deal effectively with the TV-YBLME.

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Knot Theory: From Fox 3-Colorings of Links to Yang–Baxter Homology and Khovanov Homology

Cited by 2 publications

References 29 publications

Commuting Outer Inverse-Based Solutions to the Yang–Baxter-like Matrix Equation

Commuting Outer Inverse-Based Solutions to the Yang–Baxter-like Matrix Equation

Zeroing Neural Network Approaches Based on Direct and Indirect Methods for Solving the Yang–Baxter-like Matrix Equation

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