On the minimizers of the Möbius cross energy of links. (English) Zbl 1116.58300

He, Zheng-Xu

Exp. Math. 11, No. 2, 244-248 (2002).

Summary: We give a geometric interpretation for the Euler-Lagrange equation for the Möbius cross energy of (nontrivially linked) 2-component links in the euclidean 3-space. The minimizer of this energy is conjectured to be a Hopf link of 2 round circles. We prove some elementary properties of the minimizers using the Euler-Lagrange equations. In particular, we give a rigorous proof of the fact that the minimizer is topologically a Hopf link.

MSC:

58E10	Variational problems in applications to the theory of geodesics (problems in one independent variable)
49J10	Existence theories for free problems in two or more independent variables
57M25	Knots and links in the \(3\)-sphere (MSC2010)

Keywords:

Möbius cross energy; Hopf link

References:

[1]	Abrams A., ”Circles minimize most knot energies,” (2000) · Zbl 1030.57006
[2]	DOI: 10.1016/S0166-8641(97)00211-3 · Zbl 0924.57012 · doi:10.1016/S0166-8641(97)00211-3
[3]	Cantarella J., J. Math. Physics 41 (8) pp 5615– (2000) · Zbl 1054.78500 · doi:10.1063/1.533429
[4]	Cantarella J., ”Influence of geometry and topology on helicity,” (1998)
[5]	Diao Y., ”The lower bounds of the lengths of thick knots,” (2001) · Zbl 1028.57007
[6]	Freedman M. H., Annals of Mathematics 134 (1) pp 189– (1991) · Zbl 0746.57011 · doi:10.2307/2944336
[7]	Freedman M. H., Annals of Mathematics 139 (1) pp 1– (1994) · Zbl 0817.57011 · doi:10.2307/2946626
[8]	He Z.-X., Comm, on Pure and Appl. Math. 53 (4) pp 399– (2000) · Zbl 1042.53043 · doi:10.1002/(SICI)1097-0312(200004)53:4<399::AID-CPA1>3.0.CO;2-D
[9]	Kim D., Experimental Mathematics 2 (1) pp 1– (1993) · Zbl 0818.57007 · doi:10.1080/10586458.1993.10504264
[10]	Kusner R. B., Geometric Topology (Athens, GA, 1993) pp 570– (1997)
[11]	Kusner R. B., Topology and geometry in polymer science (Minneapolis, MN, 1996) pp 67– (1998) · doi:10.1007/978-1-4612-1712-1_7
[12]	DOI: 10.1016/0040-9383(91)90010-2 · Zbl 0733.57005 · doi:10.1016/0040-9383(91)90010-2
[13]	O’Hara J., Ideal Knots (Series on Knots and Everything) 19 pp 288– (1998) · doi:10.1142/9789812796073_0016