2015/01/21 by Vassily Olegovich Manturov, Manturov, Vassily Olegovich · 1 citation
Mathematics · #05C83 #05E18 #54H20 #57M25 #57M27 #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1501.05208
openalex publication_date 2015/01/21 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
Classical knot theory deals with em diagrams and em invariants. By\nmeans of horizontal em trisecants, we construct a new theory of classical\nbraids with invariants valued in em pictures.\n These pictures are closely related to diagrams of the initial object.\n The main tool is the notion of em free k-braid group. In the simplest\ncase, for free 2-braids, the word problem and the conjugacy problem can be\nsolved by finding the minimal representative, which can be thought of as a\ngraph, and is unique, as such.\n We prove a general theorem about invariants of dynamical systems which are\nvalued in such groups and hence, in pictures.\n We describe various applications of the above theory: invariants of weavings\n(collections of skew lines in R3), and many other objects in geometry and\ntopology.\n In general, provided that for some topological objects (considered up to\nisotopy, homotopy etc) some easy axioms (coming from some dimensional\nconstraints) hold, one can construct similar dynamical systems and\npicture-valued invariants.\n