The Kauffman polynomial of order 1

Yasuyuki Miyazawa

KNOTS in WASHINGTON 10 Japan-USA; workshop in Knot Theory
January 23-30, 2000
George Washington University
Washington, DC, USA

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

Conference Homepage

The Kauffman polynomial of order 1
by
Yasuyuki Miyazawa
Yamaguchi University

In 1997, Y. Rong defined a polynomial invariant of a link which is related to the Homfly polynomial and Vassiliev invariant (of order 1). The invariant is called the first order skein (or Homfly) polynomial of a link. It satisfies the skein relations

P_{L_×}=xP_L₊+yP_{L_-}+zP_L₀ and P_{L_××}=0.

In this talk, we study a regular isotopy invariant H satisfying the following skein relations

H_{L_×}=xH_L₊+yH_{L_-}+zH_L₀+zH_{L_\infty} and H_{L_××}=0.

(In some sense, H may be thought of as the first order Kauffman polynomial of a link.) In order to do that, we will introduce a magnetic graph for a definition of the Kauffman polynomial (of order 0) of a link. Using the graph and the definition, we will define the Kauffman polynomial of order 1 for a link.

Date received: January 18, 2000