Fiber surfaces from alternating states

Abstract

In this paper we define alternating Kauffman states of links and we characterize when the induced state surface is a fiber. In addition, we give a different proof of a similar theorem of Futer, Kalfagianni and Purcell on homogeneous states.2010 Mathematics Subject Classification. 57M25, 57M15, 57M50.

State graphs and fibered state surfaces

“…Note that Girão, Nogueira, and Salgueiro also obtained a result that a particular state surface is a fiber if and only if the associated state graph is a tree [13]. However, the converse to Corollary 2.8 is not true.…”

“…These include standard diagrams of pretzel links, due to Gabai [10], and Montesinos knots, due to Hirasawa and Murasugi [17]. Previous results on fibering of state surfaces for restricted states are also along these lines [6,7,12,13]. These are the types of results that we wish to generalize.…”

“…In [7], it is shown that for some special states, fibering is also indicated immediately by the colored Jones polynomial. For other restrictions of state surfaces, the problem of fibering such state surfaces has been considered by Girão [12] and Girão, Nogueira, and Salgueiro [13]. However, we wish to find ways of classifying fibrations of all state surfaces, without restrictions on the Kaufmann state.…”