Abstract
In this chapter we describe a software code developed by the authors and capable to manipulate the topological structure of apparent contours [15].
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Notes
- 1.
See Sect. 2.5
- 2.
Most of the pictures of apparent contours in this chapter were obtained with the showcontour software, sometimes with slight additions/modifications. The command used is “showcontour <file>.morse --ge xfig --skiprtime 0.5”, it reads a Morse description of the contour (often obtained with contour printmorse) and writes a picture in xfig format.
- 3.
Of course the enclosing quotation marks ‘“’ and ‘”’ are not part of the string itself.
- 4.
Here −λ has the meaning of time.
- 5.
By inspecting the resulting apparent contour it turns out that the first occurrence of rule T is the lowest one shown in Fig. 10.7, left. Applying the second occurrence of rule T (referring to it with T:2) produces a result that is mirror-equivalent to Fig. 10.7, right, but not diffeomorphically equivalent in the sense of Definition 2.4.2.
- 6.
Not to be confused with the external region of the whole apparent contour, it is the unbounded connected component of \(\mathbb{R}^{2}\setminus C\). Occasionally we shall refer to the external region of C as the subset of the unbounded connected component of \(\mathbb{R}^{2}\setminus C\) that coincides with the region of the apparent contour adjacent to C from the outside.
- 7.
This procedure was recently suggested to us by Giovanni Paolini, Scuola Normale Superiore, Pisa, to whom we are indebted. It was first implemented in version 2.0.0 of the program. Previous versions of appcontour suffer from an imperfect canonization procedure that could lead to different regions descriptions starting from diffeomorphically equivalent apparent contours.
- 8.
As already noted in a footnote of Sect. 7.5.2, the isomorphism problem is decidable in the special case of the fundamental groups of 3-manifolds.
- 9.
The “Kinoshita-Terasaka” knot has the same property. Indeed the Conway and the Kinoshita-Terasaka knots form a “mutant” pair, they are non-equivalent knots with the same Alexander and Jones polynomials.
- 10.
As we shall see in Sect. 10.10.7 this particular syntax can be used to directly feed a Laurent ideal to the software.
- 11.
To conclude that the link cannot be split we also need to know that separately each component of the links is the unknot. That is clear from the picture, but could also be seen by concatenating the command “contour --extractcc 1 whitehead” or “contour --extractcc 2 whitehead” with “contour fg --out”.
- 12.
The elements of L can be represented by strings, a well-ordering on strings can e.g., be defined by first comparing the strings length and, in case of equal length, by making a lexicographic comparison.
- 13.
In the context of knot theory this would correspond to the sum of the two diagrams, however the result is different than the horizontal sum as apparent contours (of a tubular neighbourhood), indeed the horizontal sum of the apparent contour will apply surgery only on one of the two “parallel” arcs bounding each of the arcs of the diagrams involved in the sum operation of knots. Moreover one of the two “summands” is not in this case a knot, making the result of the sum not well defined.
- 14.
The Conway knot and its “mutant”, the Kinoshita–Terasaka knot, are the simplest nontrivial knots, in terms of number of crossing in their diagram, having trivial Alexander polynomial.
- 15.
Here is the list of equivalent presentations after each Tietze transformation: <a,b,c; AbaBcaC>, <a,b,c; BAbacabC>, <a,b,c; BBAbabcabbC>, <a,b,c; CBCBAbcabccabcb>, <a,b,c; ACBACBAbcaabcacaabcab>.
- 16.
This is not unexpected: the abelianized of the fundamental group coincides with the first homology group of the inside set E, the rank of which is the Betti number b 1 and can be computed by using the Euler–Poincaré characteristic χ = 0 of E as b 1 = b 0 + b 2 −χ where the Betti number b 0 = 1 is the number of connected components of E and the Betti number b 2 = 1 is the number of “voids” of E, which in our context is the number of connected components of ∂ E decreased by 1.
- 17.
Actually the unknotting theorem in [18, page 103] implies that indeed both the inside and outside are solid tori (in \(\mathbb{S}^{3}\)) and that they are both unknotted.
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Bellettini, G., Beorchia, V., Paolini, M., Pasquarelli, F. (2015). The Program “Appcontour”: User’s Guide. In: Shape Reconstruction from Apparent Contours. Computational Imaging and Vision, vol 44. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-45191-5_10
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