A lower bound for the number of Reidemeister moves of type III

We study the number of Reidemeister type III moves using Fox n-colorings of knot diagrams.Comment: Dedicated to Professor Louis H. Kauffman for his 60th birthda

A comparative molecular approach to mesodermal patterning in basal deuterostomes: the expression pattern of Brachyury in the enteropneust hemichordate Ptychodera flava

This work concerns the formation of mesoderm in the development of an enteropneust hemichordate, Ptychodera flava, and the expression of the Brachyury gene during this process. Brachyury expression occurs in two distinct phases. In the embryo, Brachyury is transcribed during gastrulation in the future oral and anal regions of the gut, but transcripts are no longer detected by 2 weeks of development. Brachyury expression is not detected during the 5 months of larval planktonic existence. During this time, the adult coeloms begin to develop, originating as coalescences of cells that appear to delaminate from the wall of the gut. Brachyury expression cannot be detected again until metamorphosis, when transcripts appear in the mesoderm of the adult proboscis, collar and the very posterior region of the trunk. It is also expressed in the posterior end of the gut. At no time is Brachyury expressed in the stomochord, the putative homologue of the chordate notochord. These observations illuminate the process of maximal indirect development in Ptychodera and, by comparison with patterns of Brachyury expression in the indirect development of echinoderms, their sister group, they reveal the evolutionary history of Brachyury utilization in deuterostomes

A data envelopment analysis approach to multicriteria decision problems with incomplete information

AbstractAs a decision aid for discrete multicriteria decision problems, this paper proposes a multilevel graph of alternatives to represent the ranking, to the extent that this is possible when incomplete information on weights is available under the assumption of the additive value function. To construct it, the nested decomposition of the set of alternatives is established along the lines of data envelopment analysis (DEA). A numerical example is given to illustrate a multilevel graph based on the nested decomposition and compare it with the hierarchical dominance graph based on dominance relations proposed by Park and Kim

A Theorem of Sanderson on Link Bordisms in Dimension 4

The groups of link bordism can be identified with homotopy groups via the Pontryagin-Thom construction. B.J. Sanderson computed the bordism group of 3 component surface-links using the Hilton-Milnor Theorem, and later gave a geometric interpretation of the groups in terms of intersections of Seifert hypersurfaces and their framings. In this paper, we geometrically represent every element of the bordism group uniquely by a certain standard form of a surface-link, a generalization of a Hopf link. The standard forms give rise to an inverse of Sanderson's geometrically defined invariant.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-14.abs.htm