Building links from planar diagrams

SnapPy includes the Spherogram module which allows one to create links programmatically. The graphical conventions used are summarized here.

First, here is the figure-8 knot assembled manually from four crossings, with conventions similar to those used by KnotTheory:

>>> a, b, c, d = [Crossing(x) for x in 'abcd']
>>> a[0], a[1], a[2], a[3] = c[1], d[0], b[1], b[0]
>>> b[2], b[3] = d[3], c[2]
>>> c[3], c[0] = d[2], d[1]
>>> L = Link([a,b,c,d])
>>> E = L.exterior()
>>> E.volume()
2.029883212819
>>> Manifold('4_1').is_isometric_to(E)
True

We can also give the same knot as a rational tangle:

>>> L = RationalTangle(3,5).denominator_closure()
>>> L.PD_code()
[[6, 3, 7, 4], [4, 2, 5, 1], [0, 6, 1, 5], [2, 7, 3, 0]]
>>> L.DT_code(True)
'DT[dadCDAB]'

The natural algebra of tangles shown here all works. For instance, we can build the (-2, 3, 7) pretzel knot by adding together three rational tangles:

>>> T = RationalTangle(-1, 2) + RationalTangle(1, 3) + RationalTangle(1, 7)
>>> L = T.numerator_closure()
>>> Manifold('m016').is_isometric_to(L.exterior())
True

To create the figure-8 knot as a closed braid, we first mash tangles together horizontally using “|” to make the standard braid generators; then multiplication in the braid group is just tangle multiplication:

>>> C, Id = RationalTangle(1), IdentityBraid(1)
>>> sigma_1 = C | Id
>>> x = sigma_1 = C | Id
>>> y = sigma_2_inverse = Id | -C
>>> L = (x*y*x*y).denominator_closure()
>>> Manifold('4_1').is_isometric_to(E)
True

Here’s the minimally-twisted five chain from Figure 2 of this paper:

def twisted_chain(n, k):
     T = RationalTangle(1, 2)
     m = (n+1)//2
     base = (m*[T, -T])[:n]
     tangles = base + [RationalTangle(k)]
     return sum(tangles, RationalTangle(0) ).bridge_closure()

>>> L = twisted_chain(5, -1)
>>> L.exterior().volume()
10.14941606410