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A knot can be represented with a Morse link presentation, as a combination of cups, caps and crossings (which is not uniquely determined by the knot, of course):

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Two Morse link presentations of the same knot can be related by a sequence of the following moves:

  • Swapping the order of two independent operations

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  • Pulling a cap, a cup or a crossing through two crossings (generalization of Reidemeister type 3) enter image description here
  • Cancelling out two successive crossings of alternating types (Reidemeister type 2)

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  • Twisting a cap or a cup (Reidemeister type 1)

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  • Introducing or eliminating a zig-zag

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Given Morse presentation of the unknot, it seems to me that one can always simplify it to the trivial unknot diagram without ever introducing a zig-zag (so, never introducing new strands). So the last equality could be used in the simplifying direction only. Is this fact known and if so, how does the proof go? If not, I would be interested in a counter-example.

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2 Answers 2

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I understood your question. I think it is true. First we isotope a Morse position to a bridge position without zig-zag moves. Then we have a bridge position of the trivial knot, which has been proved to be unique up to bridge isotopies by Otal. Hence we have the trivial knot diagram in a bridge position.

Otal, Jean-Pierre, Présentations en ponts du nœud trivial, C. R. Acad. Sci., Paris, Sér. I 294, 553-556 (1982). ZBL0498.57001.

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  • $\begingroup$ Thanks a lot for this pointer! I will read up on this and get back to you. It seems very promising indeed! $\endgroup$
    – pintoch
    Nov 19, 2020 at 20:19
  • $\begingroup$ It does seem to be a very elegant solution to the problem, thank you very much! I'll follow up separately to give you more context about the project. $\endgroup$
    – pintoch
    Nov 19, 2020 at 21:19
  • $\begingroup$ Otal's article can be read here: gallica.bnf.fr/ark:/12148/bpt6k5533896m/f39.item $\endgroup$
    – pintoch
    Nov 20, 2020 at 8:03
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I think this was Question 3.5 in "Thin position in the theory of classical knots" by Martin Scharlemann, and a counterexample was given by Zuapn.

Zupan, Alexander, Unexpected local minima in the width complexes for knots, Algebr. Geom. Topol. 11, No. 2, 1097-1105 (2011). ZBL1227.57015.

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  • $\begingroup$ Thanks! I am yet to fully master the notions used there, but intuitively the zig-zag rule is not the only one that can change the width of the diagram, so I have the impression that this is a slightly different problem, no? I think I can untie the example given (Figure 1, page 1101) without ever creating a new zig-zag. $\endgroup$
    – pintoch
    Nov 19, 2020 at 14:12

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