An Extended Bracket Polynomial for Virtual Knots

Louis H. Kauffman

Knots in Washington XXVIII; Follow up to Workshop on Knots and Quantum Computing (December, 2007 at UT Dallas)
February 27 - March 1, 2009
George Washington University
Washington, DC, USA

Organizers
Mieczyslaw K. Dabkowski (UTD), Jozef H. Przytycki (GWU), Vish Ramakrishna (UTD), Yongwu Rong (GWU), Alexander Shumakovitch (GWU), Kouki Taniyama (Waseda and GWU), Hao Wu (GWU)

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An Extended Bracket Polynomial for Virtual Knots
by
Louis H. Kauffman
University of Illinois at Chicago

This talk will discuss an extension of the bracket polynomial for oriented virtual knots and links that takes values in the module generated by isotopy classes of virtual 4-regular graphs over the ring of Laurent polynomials Q[A, A^-1] where Q denotes the integers. This invariant is constructed by using an oriented state expansion and keeping as much combinatorial structure in the state sum as one can. Applications of the invariant and open problems will be discussed. A special case of this invariant is the arrow polynomial of the author and Heather Dye, a variant of the Miyazawa polynomial. We will discuss applications that involve the use of both the extended bracket and the arrow polynomial.

Paper reference: arXiv:0712.2546, arXiv:0810.3858

Date received: February 14, 2009