Topology and Algebra of Bonded Knots and Braids

Ioannis Diamantis, Louis H. Kauffman and Sofia Lambropoulou ()
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Ioannis Diamantis: Department of Data Analytics and Digitalisation, School of Business and Economics, Maastricht University, P.O. Box 616, 6200 MD Maastricht, The Netherlands
Louis H. Kauffman: Department of Mathematics, Statistics and Computer Science University of Illinois at Chicago, 851 South Morgan Street, Chicago, IL 60607-7045, USA
Sofia Lambropoulou: School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Zografou Campus, GR-15780 Athens, Greece

Mathematics, 2025, vol. 13, issue 20, 1-40

Abstract: In this paper we present a detailed study of bonded knots and their related structures, integrating recent developments into a single framework. Bonded knots are classical knots endowed with embedded bonding arcs modeling physical or chemical bonds. We consider bonded knots in three categories (long, standard, and tight) according to the type of bonds, and in two categories, topological vertex and rigid vertex, according to the allowed isotopy moves, and we define invariants for each category. We then develop the theory of bonded braids , the algebraic counterpart of bonded knots. We define the bonded braid monoid , with its generators and relations, and formulate the analogues of the Alexander and Markov theorems for bonded braids in the form of L-equivalence for bonded braids. Next, we introduce enhanced bonded knots and braids , incorporating two types of bonds (attracting and repelling) corresponding to different interactions. We define the enhanced bonded braid group and show how the bonded braid monoid embeds into this group. These models capture the topology of chains with inter and intra-chain bonds and suggest new invariants for classifying biological macromolecules.

Keywords: bonded links; long bonds; tight bonds; topological vertex equivalence; rigid vertex equivalence; unplugging; tangle insertion; bonded bracket polynomial; bonded braids; bonded braid monoid; enhanced bonds; bonded braid group; bonded L-moves; bonded Alexander theorem; bonded Markov theorem (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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