The almost alternating diagrams of the trivial knot

Bankwitz characterized an alternating diagram representing the trivial knot. A non-alternating diagram is called almost alternating if one crossing change makes the diagram alternating. We characterize an almost alternaing diagram representing the trivial knot. As a corollary we determine an unknotting number one alternating knot with a property that the unknotting operation can be done on its alternating diagram.Comment: 29 pages, 18 figure

Establishment of an animal model of a pasteurized bone graft, with a preliminary analysis of muscle coverage or FGF-2 administration to the graft

<p>Abstract</p> <p>Background</p> <p>Pasteurized bone grafting is used following the excision of a bone tumor for the purpose of eliminating neoplastic cells while preserving bone-inducing ability. In the hopes of guaranteeing the most favourable results, the establishment of an animal model has been urgently awaited. In the course of establishing such a model, we made a preliminary examination of the effect of muscle coverage or fibroblast growth factor 2 (FGF-2) administration radiographically.</p> <p>Methods</p> <p>Forty pasteurized intercalary bone grafts of the Wistar rat femur treated at 60°C for 30 min were reimplanted and stabilized with an intramedullary nail (1.1 mm in diameter). Some grafts were not covered by muscle after the implantation, so that they could act as a clinical model for wide resection, and/or these were soaked with FGF-2 solution prior to implantation. The grafts were then divided into 3 groups, comprising 12 grafts with muscle-covering but without FGF-2 (MC+; FGF2-), 12 grafts without muscle-covering and without FGF-2 (MC-; FGF2-) and 16 grafts without muscle covering but with FGF-2 (MC-; FGF2+).</p> <p>Results</p> <p>At 2 weeks after grafting, the pasteurized bone model seemed to be successful in terms of eliminating living cells, including osteocytes. At 4 weeks after grafting, partial bone incorporation was observed in half the (MC+; FGF2-) cases and in half the (MC-; FGF2+) cases, but not in any of the (MC-; FGF2-) cases. At 12 weeks after grafting, bone incorporation was seen in 3 out of 4 in the (MC+; FGF2-) group (3/4: 75%) and in 3 out of 8 in the (MC-; FGF2+) group (3/8: 38%). However, most of the grafted bones without FGF-2 were absorbed in all the cases, massively, regardless of whether there had been muscle-covering (MC+; FGF2-; 4/4: 100%) or no muscle-covering (MC-; FGF2-; 4/4: 100%), while bone absorption was noted at a lower frequency (2/8: 25%) and to a lower degree in the (MC-; FGF2+) group.</p> <p>Conclusion</p> <p>In conclusion, we have established an animal pasteurized bone graft model in rats. Pasteurized bone was able to maintain bone induction ability. Despite the low number of cases in each group, the results of each group suggest that muscle-covering has an effect on bone incorporation, but that it is not able to prevent bone absorption to the pasteurized bone. However, an application of FGF-2 may have a positive effect on bone incorporation and may be able to prevent bone absorption of the graft in cases of pasteurized bone graft.</p