Invariants of knot diagrams and diagrammatic knot invariants

University dissertation from Uppsala : Acta Universitatis Upsaliensis

Abstract: Two knot diagrams describe isotopic knots if and only if they can be connected by a sequence of planar isotopies and Reidemeister moves (cusp-, self-tangency-, and triple point moves). In this dissertation classes of knot diagrams, Reidemeister moves, and relations between sequences of Reidemeister moves are investigated, in a manner inspired by V. Arnold's theory of plane curves. The local knot diagram invariants are classified, and the concept of knot diagram invariants of nite degree is introduced. Invariants of every nite degree, that jump only under triple point moves, are presented. These invariants prove that triple point moves are necessary for connecting some diagrams of isotopic knots. It is shown that there exists no non-trivial knot diagram invariants of nite degree that jump only under self-tangency moves. That is, from the view point of nite degree invariants, the self-tangency move is superfluous.The most refined topological classification of Reidemeister moves is introduced. This classification distinguishes 24 classes of moves. In particular, it distinguishes some triple point moves that only dier in the cyclic ordering in which the three branches appear on the knot. This information is vital when dealing with knot invariants defined through Gauss diagrams. An algorithm to replace any sequence of Reidemeister moves by a sequence of moves of only six out of the 24 classes is given. This gives a useful criterion to determine whether a Gauss diagram function defines a knot invariant.A graphical calculus of diagrammatic knot invariants has been developed by M. Polyak and O. Viro. M. Goussarov proved that this arrow diagram calculus provides formulas for all Vassiliev knot invariants. In this dissertation the material of Polyak and Viro's note, which contained no proofs, is presented with all proofs and details, in a self-contained form. The refined criterion for a Gauss diagram function to define a knot invariant is used to prove several formulas for invariants of knots and links, proposed by Polyak and Viro.

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