Knots and their Morse functions - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T20:31:52Z http://mathoverflow.net/feeds/question/90151 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/90151/knots-and-their-morse-functions Knots and their Morse functions muta yasushi 2012-03-03T21:54:17Z 2012-03-04T02:02:43Z <p>I interested in co-dimension 2 projections of knots. A Knot is a embedded circle in 3-space. We want to project it into 1-space. Then we use a Morse function and it appears critical points as singularities. According to the singularity theory, A knot move is made by a surface knot and a projection. For example, Reidemeister moves are considered as neighborhoods of singularities of surfaces in 3-space. The surfaces in 3-space is considered as a co-dimension 1 projection of a surface knot which is a embedded surface into 4-space and represent a knot isotopy. Similarly, we want to consider a co-domension 2 projection of a surface knot. Then it appears folds and cusps as singularities. The neighborhoods of a fold and a cusp may make moves. For recostructing from a projection into 1-space of a knot, we replace a knot as a set of braids with critical point's information. A Morse functions of a knot make intervals between critical points. The pre-image of the interbal is a braid with critical point's information. To neighborhoods of a fold and a cusp, we add the information of braids with critical point's information, that is maybe new knot moves.</p> <p>Question</p> <p>Can we make new knot moves for Morse functions like above? Moreover, is it well-known the new move?</p> <p>Thank you for your considerations.</p> http://mathoverflow.net/questions/90151/knots-and-their-morse-functions/90173#90173 Answer by Scott Carter for Knots and their Morse functions Scott Carter 2012-03-04T02:02:43Z 2012-03-04T02:02:43Z <p>The answer to your question is a qualified, yes. The full reference is <a href="http://arxiv.org/pdf/math/9912016.pdf" rel="nofollow"> this article </a> by Cooper, Mond and Wit Atique. In it they describe complex multi-germs of functions. This singularity theory is an ingredient in any approach to the Reidemeister moves for higher dimensional knots (beyond 2-knots in 4-space). </p> <p>For knotted surfaces, the original work of Roseman and independently Homme/Nagase gives the 7 moves that are necessary to move a knotted surface around. The best movie move version of Roseman's theorem is work of mine with Joachim Rieger and Masahico Saito. The non-pay wall version is <a href="http://www.southalabama.edu/mathstat/personal_pages/carter/ca_ri_sa.pdf" rel="nofollow"> here </a>. But see also our book <a href="http://www.amazon.com/Knotted-Surfaces-Diagrams-Mathematical-Monographs/dp/0821805932/ref=sr_1_1?ie=UTF8&amp;qid=1330825498&amp;sr=8-1" rel="nofollow"> Knotted Surfaces and their diagrams </a> or <a href="http://www.amazon.com/Surfaces-4-Space-Encyclopaedia-Mathematical-Sciences/dp/3642059139/ref=ntt_at_ep_dpt_3" rel="nofollow"> Surfaces in $4$-space.</a> </p> <p>To get a good understanding of cusps and how they behave, I suggest <a href="http://www.amazon.com/Excursion-Diagrammatic-Algebra-Turning-Everything/dp/9814374490/ref=ntt_at_ep_dpt_5" rel="nofollow"> this recent book </a>. So having shamelessly promoted my work on this let me describe a little on how to approach the general problem in higher dimensions.</p> <p>Staring from the Reidemeister moves of an $n$-manifold embedded in $(n+2)$-space (and considering their projection in $(n+1)$-space). We use these to construct the singularities of $(n+1)$-manifolds in $(n+3)$ space. These singularities together with the Morse critical points of the $(n+1)$-manifold are the ingredients used to create the knotting. To determine the Reidemeister moves of $(n+1)$-manifolds in $(n+2)$-space one first posits that the lower dimensional R-moves are invertible on both sides. (So for example a there is a type-II saddle and a type-II bubble move). One posits a high dimensional version of the R-III move. In dim. 4 this is the tetrahedral move. In general it corresponds to moving the hyperplane $\sum_j x_j = 1$ across the coordinate planes $x_j=0$ to the plane $\sum_j x_j=-1$. The remaining moves are going to involve the branch points and the analogues of the R-III move. Specifically, branch points can be moved through transverse sheets. This is where Cooper-Mond-Wit Atique is needed. </p> <p>Finally, I think one <em>should</em> be able to construct a higher dimensionsal movie move theorem by means of examining the interactions between the lower dimensional R-moves, and the critical points of the various strata. As long as you can determine that singularities are codimension 1 type, then you have all the moves.</p>