The Andreev-Thurston theorem states that for any triangulation of a closed orientable surface Σ g of genus g which is covered by a simple graph in the universal cover, there exists a unique metric of curvature 1, 0 or −1 on the surface depending on whether g = 0, 1 or ≥ 2 such that the surface with this metric admits a circle packing with combinatorics given by the triangulation. Furthermore, the circle packing is essentially rigid, that is, unique up to conformal automorphisms of the surface isotopic to the identity. In this paper, we consider projective structures on the surface Σ g where circle packings are also defined. We show that the space of projective structures on a surface of genus g ≥ 2 which admits a circle packing by one circle is homeomorphic to R 6g−6 and furthermore that the circle packing is rigid on such surfaces. 1991 Mathematics Subject Classification. Primary 52C15; Secondary 30F99, 57M50.

We give an upper bound on the denominators of numerical boundary slopes and an upper bound on the differences between two numerical boundary slopes for Montesinos knots.

It is shown that the diameter of the boundary slope set is bounded from above by the twice of the minimal crossing number for a Montesinos knot.

Goblet cell carcinoids are rare neoplasms that predominantly occur in the appendix. In this report we present a case of goblet cell carcinoid of the appendix. A 58-year-old male patient complaining of pain in the right lower quadrant was diagnosed with acute appendicitis and underwent an appendectomy. Histological examination of the resected appendix revealed goblet cell carcinoid. Infiltration of tumor cells beyond the appendix was observed and the surgically resected margin was positive for tumor cells. Carcinoembryonic antigen (CEA) was diffusely detected by immunohistochemistry, and cytokeratin 20, neuron-specific enolase (NSE), chromogranin A and serotonin were focally observed in the tumor cells. The expression of beta-catenin and E-cadherin was investigated to compare with that of typical rectal carcinoids (n = 3) and colon adenocarcinomas (n = 3). In normal colonic and rectal mucosae, beta-catenin and E-cadherin stained positive on the plasma membrane. In the case reported here, beta-catenin showed a preserved expression on the plasma membrane of goblet cell carcinoid; a pattern similar to typical carcinoids rather than to adenocarcinomas. However, E-cadherin demonstrated a reduced expression on the plasma membrane of the tumor cells. This staining pattern was identical to those both of carcinoids and of adenocarcinomas. These findings suggest the possibility that, in some cases, the adherens junctions of goblet cell carcinoids are similar to those of typical carcinoids rather than to those of adenocarcinomas.

a r t i c l e i n f o a b s t r a c t MSC: 57M25 Keywords: Crosscap number Pretzel knotThe crosscap number of a knot in the 3-sphere is defined as the minimal first Betti number of non-orientable surfaces bounded by the knot. In this paper, we determine the crosscap numbers of a large class of pretzel knots. The key ingredient to obtain the result is the algorithm of enumerating all essential surfaces for Montesinos knots developed by Hatcher and Oertel.