ar X iv : m at h / 99 07 10 4 v 1 [ m at h . G T ] 1 5 Ju l 1 99 9 Orderings of mapping class groups after Thurston

ar X iv : m at h / 99 07 08 5 v 1 [ m at h . G R ] 1 3 Ju l 1 99 9 LOOPS AND SEMIDIRECT PRODUCTS

ar X iv : m at h / 99 07 18 5 v 1 [ m at h . M G ] 2 9 Ju l 1 99 9 Higher dimensional flexible polyhedra ∗

It is show that sphere homeomorphic flexible polyhedra (with self intersections) do really exist in n-dimensional Euclidean, Lobachevskij and spherical spaces for each n ≥ 3. 1991 Mathematics Subject Classification: 52C25, 52B11.

ar X iv : m at h / 99 05 15 4 v 1 [ m at h . G T ] 2 5 M ay 1 99 9 SUBSPACES OF KNOT SPACES

The inclusion of the space of all knots of a prescribed writhe in a particular isotopy class into the space of all knots in that isotopy class is a weak homotopy equivalence.

ar X iv : m at h / 99 06 19 7 v 1 [ m at h . G T ] 2 8 Ju n 19 99 The size of spanning disks for polygonal knots .

For each integer n ≥ 1 we construct a closed unknotted Piecewise Linear curve Kn in R 3 having less than 11n edges with the property that any Piecewise Linear triangluated disk spanning the curve contains at least 2 triangles.

ar X iv : m at h / 99 07 08 5 v 2 [ m at h . G R ] 1 7 D ec 1 99 9 LOOPS AND SEMIDIRECT PRODUCTS