We present formulae for computing the Yamada polynomial of spatial graphs obtained by replacing edges of plane graphs, such as cycle-graphs, theta-graphs, and bouquet-graphs, by spatial parts. As a corollary, it is shown that zeros of Yamada polynomials of some series of spatial graphs are dense in a certain region in the complex plane, described by a system of inequalities. Also, the relation between Yamada polynomial of graphs and the chain polynomial of edge-labelled graphs is obtained.

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This paper is concerned with the crosscap numbers of Montesinos knots. For a family of Montesinos knots, we give a lower bound of [Formula: see text] among all essential surfaces [Formula: see text] embedded in the exterior, where [Formula: see text] denotes the ratio of negative Euler characteristic of the surface and the number of sheets. From this result, we determine the crosscap numbers of a family of Montesinos knots. Our method relies on the algorithm of enumerating all essential surfaces for Montesinos knots given by Hatcher and Oertel.

We survey the construction and properties of the Yamada polynomial of spatial graphs and present the Yamada polynomial formulae for some classes of graphs. Then we construct an infinite family of spatial graphs for which roots of Yamada polynomials are dense in the complex plane. It is well known due S. Kinoshita [14,15], that the Alexander ideal and Alexander polynomial are invariants of spatial graphs which are determined by the fundamental groups of the complements of spatial graphs.In 1989, S. Yamada [24] introduced Yamada polynomial of spatial graphs in R 3 . It is an concise and useful ambient isotopy invariant for graphs with maximal degree less than four. There are many interesting results on Yamada polynomial and its generalizations. J. Murakami [17] investigated the two-variable extension Z S of the Yamada polynomial and gave an invariant related to the HOMFLY polynomial. In 1994, the crossing number of spatial graphs in terms of the reduced degree of Yamada polynomial has been studied by T. Motohashi, Y. Ohyama and K. Taniyama [19]. In 1996, A. Dobrynin and A. Vesnin [7] studied properties of the Yamada polynomial of spatial graphs. For any graph G, V. Vershinin and A. Vesnin [23] defined bigraded cohomology groups whose graded Euler characteristic is a multiple of the Yamada polynomial of G. Another invariant of spatial graphs associated with U q (sl(2, C)) was introduced 2010 Mathematics Subject Classification. Primary 57M15; Secondary 05C31.