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  • Knot Group and the Unknot - Mathematics Stack Exchange
    This means that the boundary torus is not incompressible So there is a simple closed curve on the torus that bounds a disk into the complement Furthermore, the only such curve which is null-homologous is the longitude Therefore, the longitude is null-homotopic and by the disk theorem bounds an embedded disk Thus the knot is the unknot
  • Why the Alexander polynomial of the unknot (trivial knot ) is the . . .
    The obvious presentation of the unknot has no crossings, so you're taking the determinant of a $0\times0$ matrix, which equals $1$ The Alexander polynomial $\Delta$ is a reparameterization of the Alexander-Conway polynomial $\nabla$ , which is defined by a sort of "recurrence relation" whose base case is $\nabla(\textrm{unknot})=1$
  • Is unknot a composite knot? - Mathematics Stack Exchange
    When I was reading in introduction chapter in composite knot subchapter, I came up with idea that if we somewhat composite 2 or more factor knots each another under rules of composition, we can end up with unknot knot, that is you composite knots in such a way that composition of these knot remove each other to become unknot knot
  • Prove that every knot diagram with two crossings is equivalent to the . . .
    The question I am looking to answer now: Prove that every knot diagram with two crossings is equivalent to the unknot So far, I have drawn four possible knot diagrams and shown they are equivalent to the unknot, but I am unsure how to proceed
  • Knot group is $\\mathbb{Z}$ iff $K$ is the unknot
    I've come across the fact that the only knot whose knot group is isomorphic to $\mathbb{Z}$ is the unknot;
  • geometric topology - Unknot is fibred - Mathematics Stack Exchange
    If you really want to think about this in the standard unknot in $\Bbb R^3$, it starts with the obvious disc, then the disc moves upward to look like a spherical cap; eventually it looks like most of a very large sphere with a small cap below the unknot removed; eventually it becomes the complement of the disc in the xy-plane (still a disc once
  • Is the Trefoil knot isotopic to the unknot (torus) in ambient
    $\begingroup$ @axel22 The trefoil knot picture in Wikipedia is misleading You can shrink the diameter of the "thread" making up that knot to zero to see that this should really be seen as a 1-dimensional knot: there's only one degree of freedom in moving along the knot itself
  • Do there exist non-trivial knots whose Jones polynomial is a unit?
    $\begingroup$ Note that Khovanov homology (which is a refinement of the Jones polynomial), does detect the unknot $\endgroup$ – Moishe Kohan Commented Apr 25, 2020 at 15:32
  • Why are all knots trivial in 4D? - Mathematics Stack Exchange
    Any knot has crossings, and changing enough of them from over to under or vice versa will give us the unknot (The minimum number of changes is known as the unknotting number ) So to show that we can always get the unknot, all we have to do is show that we can change any crossing we want
  • differential topology - Unknot: embedded vs immersed bounding disk . . .
    Unknot: embedded vs immersed bounding disk Ask Question Asked 5 years, 2 months ago Modified 5 years, 2





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