The Number of Triangles Needed to Span a Polygon Embedded in R^d

Given a closed polygon P having n edges, embedded in R^d, we give upper and lower bounds for the minimal number of triangles t needed to form a triangulated PL surface in R^d having P as its geometric boundary. The most interesting case is dimension 3, where the polygon may be knotted. We use the Seifert suface construction to show there always exists an embedded surface requiring at most 7n^2 triangles. We complement this result by showing there are polygons in R^3 for which any embedded surface requires at least 1/2n^2 - O(n) triangles. In dimension 2 only n-2 triangles are needed, and in dimensions 5 or more there exists an embedded surface requiring at most n triangles. In dimension 4 we obtain a partial answer, with an O(n^2) upper bound for embedded surfaces, and a construction of an immersed disk requiring at most 3n triangles. These results can be interpreted as giving qualitiative discrete analogues of the isoperimetric inequality for piecewise linear manifolds.Comment: 16 pages, 4 figures. This paper is a retitled, revised version of math.GT/020217

The number of Reidemeister Moves Needed for Unknotting

There is a positive constant $c_1$ such that for any diagram $D$ representing the unknot, there is a sequence of at most $2^{c_1 n}$ Reidemeister moves that will convert it to a trivial knot diagram, $n$ is the number of crossings in $D$. A similar result holds for elementary moves on a polygonal knot $K$ embedded in the 1-skeleton of the interior of a compact, orientable, triangulated $PL$ 3-manifold $M$. There is a positive constant $c_2$ such that for each $t \geq 1$, if $M$ consists of $t$ tetrahedra, and $K$ is unknotted, then there is a sequence of at most $2^{c_2 t}$ elementary moves in $M$ which transforms $K$ to a triangle contained inside one tetrahedron of $M$. We obtain explicit values for $c_1$ and $c_2$.Comment: 48 pages, 14 figure

Area Inequalities for Embedded Disks Spanning Unknotted Curves

We show that a smooth unknotted curve in R^3 satisfies an isoperimetric inequality that bounds the area of an embedded disk spanning the curve in terms of two parameters: the length L of the curve and the thickness r (maximal radius of an embedded tubular neighborhood) of the curve. For fixed length, the expression giving the upper bound on the area grows exponentially in 1/r^2. In the direction of lower bounds, we give a sequence of length one curves with r approaching 0 for which the area of any spanning disk is bounded from below by a function that grows exponentially with 1/r. In particular, given any constant A, there is a smooth, unknotted length one curve for which the area of a smallest embedded spanning disk is greater than A.Comment: 31 pages, 5 figure

Topological and physical knot theory are distinct

Physical knots and links are one-dimensional submanifolds of R^3 with fixed length and thickness. We show that isotopy classes in this category can differ from those of classical knot and link theory. In particular we exhibit a Gordian Split Link, a two component link that is split in the classical theory but cannot be split with a physical isotopy.Comment: 8 page, 6 figure

Stepwise transition from deglacial/Early Holocene to modern-like conditions in the eastern Fram Strait, sub-Arctic north, inferred from planktic foraminifer fauna and sea surface temperatures

EGU2012-4750 The heat content of the Arctic Ocean is mainly controlled by the inflow of north-heading warm and saline Atlantic Water through eastern Fram Strait. The eastern Fram Strait is therefore ice-free all year, opposite to its perennially ice-covered western part where large amounts of Arctic sea ice are exported year-round to the Nordic Seas. The Early and Mid-Holocene phases (ca 12 to 5 cal ka BP) in the (sub-)Arctic have been especially marked not only by high summer insolation but also by rising sea level and the final disintegration of large ice sheets that had been established during the preceding glacial phase. Two sediment cores with multidecadal resolution from the Western Svalbard margin have been investigated for its planktic foraminiferal distribution, sea surface temperatures, planktic and benthic stable isotope ratios, and lithological parameters to derive information on the Holocene variability of the heat transport to the Arctic Ocean and related fluctuations of the marginal ice zone in the eastern Fram Strait. Planktic foraminifer fauna and a summer sea surface temperature reconstruction based on the modern analogue technique imply a stepwise transition from deglacial/Early Holocene to modern-like conditions in the eastern Fram Strait. Repeated short-term advances of the sea ice margin accompanied the generally strong heat transport to the Arctic Ocean during the Early to Mid-Holocene. Consistent with the decreasing solar insolation, cooler (sub-)surface conditions established after ca 5 cal ka BP most likely related to both a weakening of the Atlantic Water inflow and strong export of Arctic sea ice through Fram Strait. The Late Holocene Neoglacial phase was characterized by high contents of ice-rafted material and dominance of the cold water-indicating planktic foraminifer species Neogloboquadrina pachyderma. Cool Late Holocene conditions are reversed by a strong warming event likely caused by a significant strengthening of Atlantic heat advection to the Arctic during the present, anthropogenically influenced period

Pole Dancing: 3D Morphs for Tree Drawings

We study the question whether a crossing-free 3D morph between two straight-line drawings of an $n$-vertex tree can be constructed consisting of a small number of linear morphing steps. We look both at the case in which the two given drawings are two-dimensional and at the one in which they are three-dimensional. In the former setting we prove that a crossing-free 3D morph always exists with $O(\log n)$ steps, while for the latter $\Theta(n)$ steps are always sufficient and sometimes necessary.Comment: Appears in the Proceedings of the 26th International Symposium on Graph Drawing and Network Visualization (GD 2018

Gender, war and militarism: making and questioning the links

The gender dynamics of militarism have traditionally been seen as straightforward, given the cultural mythologies of warfare and the disciplining of ‘masculinity’ that occurs in the training and use of men's capacity for violence in the armed services. However, women's relation to both war and peace has been varied and complex. It is women who have often been most prominent in working for peace, although there are no necessary links between women and opposition to militarism. In addition, more women than ever are serving in many of today's armies, with feminists rather uncertain on how to relate to this phenomenon. In this article, I explore some of the complexities of applying gender analyses to militarism and peace work in sites of conflict today, looking most closely at the Israeli feminist group, New Profile, and their insistence upon the costs of the militarized nature of Israeli society. They expose the very permeable boundaries between the military and civil society, as violence seeps into the fears and practices of everyday life in Israel. I place their work in the context of broader feminist analysis offered by researchers such as Cynthia Enloe and Cynthia Cockburn, who have for decades been writing about the ‘masculinist’ postures and practices of warfare, as well as the situation of women caught up in them. Finally, I suggest that rethinking the gendered nature of warfare must also encompass the costs of war to men, whose fundamental vulnerability to psychological abuse and physical injury is often downplayed, whether in mainstream accounts of warfare or in more specific gender analysis. Feminists need to pay careful attention to masculinity and its fragmentations in addressing the topic of gender, war and militarism