The main aim of this work is to develop a method of constructing higher Hamiltonians of quantum integrable systems associated with the solution of the Zamolodchikov tetrahedral equation. As opposed ...to the result of V.V. Bazhanov and S.M. Sergeev the approach presented here is effective for generic solutions of the tetrahedral equation without spectral parameter. In a sense, this result is a two-dimensional generalization of the method by J.-M. Maillet. The work is a part of the project relating the tetrahedral equation with the quasi-invariants of 2-knots. ProQuest: ... denotes formulae omitted.
Molecular knots in biology and chemistry Lim, Nicole C H; Jackson, Sophie E
Journal of physics. Condensed matter,
09/2015, Volume:
27, Issue:
35
Journal Article
Peer reviewed
Open access
Knots and entanglements are ubiquitous. Beyond their aesthetic appeal, these fascinating topological entities can be either useful or cumbersome. In recent decades, the importance and prevalence of ...molecular knots have been increasingly recognised by scientists from different disciplines. In this review, we provide an overview on the various molecular knots found in naturally occurring biological systems (DNA, RNA and proteins), and those created by synthetic chemists. We discuss the current knowledge in these fields, including recent developments in experimental and, in some cases, computational studies which are beginning to shed light into the complex interplay between the structure, formation and properties of these topologically intricate molecules.
We discuss Vassiliev invariants for virtual knots, expanding upon the theory of quantum virtual knot invariants developed in 1. In particular, following the theory of quantum invariants we work with ...‘rotational’ virtual knots. We define chord diagrams, weight systems, and give examples of Lie algebra weight systems of rotational virtual knots. We end with a discussion of extended quantum invariants, which capture information that standard quantum invariants of rotational virtuals cannot.
•Description of chord diagrams in the setting of rotational virtual knots.•Construction of Lie algebra weight systems of rotational virtual knots.•Generalization of quantum invariants of rotational virtual knots.
The computer artificial intelligence system AlphaFold has recently predicted previously unknown three‐dimensional structures of thousands of proteins. Focusing on the subset with high‐confidence ...scores, we algorithmically analyze these predictions for cases where the protein backbone exhibits rare topological complexity, that is, knotting. Amongst others, we discovered a 71‐knot, the most topologically complex knot ever found in a protein, as well several six‐crossing composite knots comprised of two methyltransferase or carbonic anhydrase domains, each containing a simple trefoil knot. These deeply embedded composite knots occur evidently by gene duplication and interconnection of knotted dimers. Finally, we report two new five‐crossing knots including the first 51‐knot. Our list of analyzed structures forms the basis for future experimental studies to confirm these novel‐knotted topologies and to explore their complex folding mechanisms.
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