Achirality of Knots

M. Azram

KNOTS in WASHINGTON XV (2nd Japan-USA Workshop in Knot Theory)
January 10-15, 2003
George Washington University and Johns Hopkins University
Washington, DC and Baltimore, MD, USA

Organizers
Kazuaki Kobayashi, Jozef H. Przytycki, Yongwu Rong, Shin-ichi Suzuki, Kouki Taniyama, Tatsuya Tsukamoto, Akira Yasuhara

Conference Homepage

Achirality of Knots
by
M. Azram
IIUM, Kuala Lumpur 53100, Malaysia

A theoretic and diagrammatic relationship between knots and planar graphs has been established. It has been shown that in reduced alternating achiral knots, number of black regions is same as the number of white regions and consequently, W(K)=0 iff B = W is a necessary condition for reduced alternating knot to be achiral. It has been proved that if a reduced alternating achiral knot has p number of black regions then it has 2(p-1) crossings. This relationship enabled us to establish that the number of crossings and regions and hence, the number of vertices, edges and faces in the corresponding LR-Graphs are invariant. It has also been established that the number of vertices, edges and regions in LR-Graphs corresponding to black and white regions of reduced alternating achiral knot are same. A new way of constructing a reduced alternating achiral knot (linked link) has been suggested. Establishment of new but pivotal moves such as R*-move , 2(\pi)-twist and (\pi)-twist enabled us to prove that the black regions can be changed into white regions via Reidemeister moves. Consequently, the equivalence of the companion graphs, necessary and sufficient conditions for a reduced alternating knot to be achiral has been established.

Date received: September 7, 2002