Abstract:
In the present paper, we suggest a new combinatorial approach to knot theory based on embeddings of knots
and links into a union of three half-planes with the same boundary. The restriction of the number of pages to three (or any other number ⩾3) provides a convenient way to encode links by words in a finite alphabet. For those words, we give a finite set of local changes that realizes the equivalence of links by analogy with the Reidemeister moves for planar link diagrams.
Citation:
I. A. Dynnikov, “Three-Page Approach to Knot Theory. Encoding and Local Moves”, Funktsional. Anal. i Prilozhen., 33:4 (1999), 25–37; Funct. Anal. Appl., 33:4 (1999), 260–269
This publication is cited in the following 12 articles:
Morozov A., Smirnov A., “Chern–Simons theory in the temporal gauge and knot invariants through the universal quantum R-matrix”, Nuclear Phys B, 835:3 (2010), 284–313
Kearton, C, “All 2-dimensional links in 4-space live inside a universal 3-dimensional polyhedron”, Algebraic and Geometric Topology, 8:3 (2008), 1223
Kurlin, V, “Three-page encoding and complexity theory for spatial graphs”, Journal of Knot Theory and Its Ramifications, 16:1 (2007), 59
Dujmovic, V, “Stacks, queues and tracks: Layouts of graph subdivisions”, Discrete Mathematics and Theoretical Computer Science, 7:1 (2005), 155
V. V. Vershinin, V. A. Kurlin, “Three-Page Embeddings of Singular Knots”, Funct. Anal. Appl., 38:1 (2004), 14–27
V. A. Kurlin, “Basic embeddings of graphs and Dynnikov's three-page embedding method”, Russian Math. Surveys, 58:2 (2003), 372–374
I. A. Dynnikov, “Recognition algorithms in knot theory”, Russian Math. Surveys, 58:6 (2003), 1093–1139
Andreeva M.V., Dynnikov I.A., Polthier K., “A mathematical webservice for recognizing the unknot”, Mathematical Software, Proceedings, 2002, 201–207
Dynnikov, IA, “A new way to represent links. One-dimensional formalism and untangling technology”, Acta Applicandae Mathematicae, 69:3 (2001), 243
V. A. Kurlin, “Dynnikov Three-Page Diagrams of Spatial 3-Valent Graphs”, Funct. Anal. Appl., 35:3 (2001), 230–233
I. A. Dynnikov, “Three-Page Approach to Knot Theory. Universal Semigroup”, Funct. Anal. Appl., 34:1 (2000), 24–32
I. A. Dynnikov, “Finitely Presented Groups and Semigroups in Knot Theory”, Proc. Steklov Inst. Math., 231 (2000), 220–237
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