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Revisiting an Extended Bracket Polynomial for Virtual Knots
by
Louis H Kauffman
University of Illinois at Chicago
We consider a bracket expansion where one replaces an oriented crossing with combination of a smoothed crossing (s) and a disoriented smoothed crossing (dsc). The dsc has the appearance of two cusps meeting one another. One cusp has arrows pointing toward one another. One cusp has arrows pointing away from each other. The two cusps are said to be paired if they are in the form of the dcc. Paired cusps form a species of decorated 4-valent vertex. We give a set of graphical replacements for certain configurations of paired cusps so that the bracket state expansion based on < K+ > = A < Ks > + A-1 < Kdsc > is an invariant of regular isotopy where the graphs that appear in the expansion are taken up to the equivalence relation generated by the graphical replacements. A special case of this invariant is the arrow polynomial, where the cusps do not have to be paired. We discuss the differences between this extended invariant and the arrow polynomial. This discussion forms the basis for further technical discussions about errors (discovered by Sergei Chmutov and Bejamin OConnor) in a previous paper by the author where he erroneously postulated a specific ordered reduction procedure for these graphs based on the above mentioned graphical moves. One can formulate a basic invariant using an equivalence relation rather than a reduction procedure and then start the investigation again in this new way.
Date received: April 26, 2017
Copyright © 2017 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cbof-26.