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Results tagged with knot-theory
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user 23571
Knot theory is dealing with embedding of curves in manifolds of dimension 3. A knot is a single circle embedded in the affine space of dimension 3 as a smooth curve not crossing itself. Many knot invariants are known and can be used to distinguish knots.
2
votes
How can i change 8_19 to (3,4)-torus knot K(3,4)?
Here is an explicit move that transforms the diagram shown into the standard projection of the torus knot. Consider the upper triangle in the diagram, and label the three crossings at its vertices as …
- 20k
6
votes
Accepted
Essential surfaces in the Exterior of Montesinos knots
Most Montesinos knots and links have closed incompressible surfaces in their complements. This was shown by Ulrich Oertel in the paper "Closed incompressible surfaces in complements of star links", Pa …
- 20k
15
votes
Satellite knot example
If by the symmetry group of a knot you mean the group of isometries of $S^3$ leaving the knot invariant, then this can only be cyclic or dihedral, apart from the special case of torus knots which can …
- 20k
10
votes
Computing the structure of the group completion of an abelian monoid, how hard can it be?
An ultra-classical example: the failure of unique factorization in algebraic number fields. Here one looks at the multiplicative monoid of nonzero algebraic integers in a finite extension field of $\m …
- 20k
24
votes
Reference for a fact (?) on homeomorphic knot complements
This is a question that I remember worrying about when I first started learning about knot theory. Older books have a tendency to skim over this point rather lightly, perhaps because the resolution of …
- 20k