Monotonic simplification of arc-presentations of some satellite knots

Kazantsev, Aleksander Dmitrievich

doi:10.1070/rm2010v065n04abeh004697

Cited by 1 publication

(2 citation statements)

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“…The Kazantsev knot A rather complicated non-trivial knot we treat comes from [20] and is shown in Fig. 17.…”

Section: Experimental Evidencementioning

confidence: 99%

“…Finally, we will turn to non-trivial (and even composite) knots, showing that our reduced algorithm is also successful (and extremely efficient) in bringing their diagrams to a minimal crossing number status. We single out here the Kazantsev knot diagram [20], with 23 crossings, which is intractable using only the Reidemeister moves, while our procedure P monotonically reduces it to its minimal status with 17 crossings via 6 moves of type Z 3 (one of which is horizontal). To conclude we will provide two examples of diagrams of non-trivial knots that require more than one consecutive horizontal Z 3 move to reach a minimal status.…”

Section: Technical Notesmentioning

confidence: 99%

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Algorithmic simplification of knot diagrams: New moves and experiments

Petronio

Zanellati²

2016

J. Knot Theory Ramifications

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This note has an experimental nature and contains no new theorems.We introduce certain moves for classical knot diagrams that for all the very many examples we have tested them on give a monotonic complete simplification. A complete simplification of a knot diagram D is a sequence of moves that transform D into a diagram D ′ with the minimal possible number of crossings for the isotopy class of the knot represented by D. The simplification is monotonic if the number of crossings never increases along the sequence. Our moves are certain Z 1,2,3 generalizing the classical Reidemeister moves R 1,2,3 , and another one C (together with a variant C) aimed at detecting whether a knot diagram can be viewed as a connected sum of two easier ones.We present an accurate description of the moves and several results of our implementation of the simplification procedure based on them, publicly available on the web. MSC (2010): 57M25.Keywords: Knot diagram, move, simplification. This paper describes constructions and experimental results originally due to the second named author, that were formalized, put into context, and generalized in collaboration with the first named author. No new theorem is proved.We introduce new combinatorial moves Z 1,2,3 on knot diagrams, together with another move C and a variant C of C, that perform very well in the task of simplifying diagrams. Namely, as we have checked by implementing the moves in [29] and applying them to a wealth of examples, the moves are * Partially supported by the Italian FIRB project "Geometry and topology of lowdimensional manifolds" RBFR10GHHH. The moves Z 1,2,3 extend the classical Reidemeister moves R 1,2,3 , while C and C are aimed at detecting whether a diagram can be viewed as the connected sum of two easier ones. In practice, the simplification procedure based on the moves Z 1,2,3 only is already quite powerful, since it allows for instance to monotonically untangle most of the known hard diagrams of the unknot. However the moves Z 1,2,3 only are not sufficient in general, as an example provided to us by the referee shows, but this example is easily dealt with using the move C. We do not know whether the use of all the moves Z 1,2,3 , C and C allows a monotonic complete simplification of any knot diagram, but if this were the case one would have an algorihtm to compute the crossing number of a knot, and in particular to detect knottedness.In the history of knot theory quite some energy has been devoted to the problem of effectively detecting whether or not a knot diagram represents the unknot, and more generally of computing the crossing number of a link starting from an arbitrary diagram representing it. The point here is that, using only Reidemeister moves, one may have to temporarily increase the number of crossings before reducing the link to a minimal crossing diagram. This phenomenon occurs both for the unknot and for more general knots and links, as explained below. Kazantsev [20] for a further development), but apparently not a method for calcula...

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“…The Kazantsev knot A rather complicated non-trivial knot we treat comes from [20] and is shown in Fig. 17.…”

Section: Experimental Evidencementioning

confidence: 99%

Section: Technical Notesmentioning

confidence: 99%