Möbius energy of thick knots

Möbius Energy of Thick Knots

The Möbius energy of a knot is an energy functional for smooth curves based on an idea of self-repelling. If a knot has a thick tubular neighborhood, we would intuitively expect the energy to be low. In this paper, we give explicit bounds for energy in terms of the ropelength of the knot, i.e. the ratio of the length of a thickest tube to its radius.

Torus Knots Extremizing the Möbius Energy

This work was accomplished while the authors were at the Institute for Advanced Study, and was partly supported by an NSF Postdoctoral Fellowship. Using the principle of symmetric criticality [Palais 1979], we construct torus knots and links that extremize the Möbiusinvariant energy introduced by O’Hara [1991] and Freedman, He and Wang [1993]. The critical energies are explicitly computable usi...

H-thick Knots in Khovanov Homology

We study Khovanov homology classes which have state cycle representatives, and examine how they interact with Jacobsson homomorphisms and Lee’s map Φ. As an application, we describe a general procedure, quasipositive modification, for constructing H-thick knots in rational Khovanov homology. Moreover, we show that specific families of such knots cannot be detected by Khovanov’s thickness criter...

Energy and Length of Knots

Knots are idealized 1-dimensional loops that tangle themselves in 3-space. They have been studied, for more than 100 years, primarily as abstract mathematical objects even though the original interest in the subject seems to be based in physics. There is now interest in reinvesting the mathematical abstractions with physical-like properties such as thickness L] Si] DEJ] St] LSDR] or self-repell...

On the Minimizers of the Möbius Cross Energy of Links