On the torsional energy of torus knots under infinitesimal bending

Abstract

The article deals with the infinitesimal bending theory application to the knots theory. The impact of infinitesimal bending on the torsional energy at torus knots is considered, and the results show that it is not stationary under infinitesimal bending. The torsional energy variation is determined as well. We prove that there is no infinitesimal bending field that leaves torus curves on the torus. Besides, we define an infinitesimal bending field that does not tear the torus knots while bending. Having in mind the importance of visualization in the infinitesimal bending theory, we observed infinitesimal bending of a curve in that field using independently developed software. The graphs we obtained are presented in the paper and the torus knots are coloured according to their torsional energy. We calculated the numerical value of torsional energy under infinitesimal bending and, finally, the results are discussed using convenient specific examples.