def from_dowker_code(self, code):
"""
Build a knot from a Dowker-Thistlethwaite code.
The Dowker-Thistlethwaite code of a knot diagram is defined as follows.
Start following the knot diagram at some regular point. Label the
crossings by a number (starting from number 1) in the order in
which they are met. At the end, every crossing gets numbered
twice, once by an even number and once by an odd number. When
meeting an over-crossing with even number, use instead the
negative of this even number as label.
Then the set of crossings gives a set of pairs (odd,
even). Sort this set according to the odd component, and then
keep only the even components in the same order. This is the
Dowker-Thistlethwaite code.
INPUT:
a list of signed even numbers, the Dowker-Thistlethwaite code of a knot
OUTPUT:
a knot
EXAMPLES::
sage: W = Knots()
sage: K1 = W.from_dowker_code([8,10,2,12,4,6])
sage: K1.dowker_notation()
[(5, 2), (9, 4), (11, 6), (1, 8), (3, 10), (7, 12)]
sage: W.from_dowker_code([6,10,14,12,16,2,18,4,8])
Knot represented by 9 crossings
sage: W.from_dowker_code([4,8,10,-14,2,-16,-18,-6,-12])
Knot represented by 9 crossings
sage: K3 = W.from_dowker_code([6,-12,2,8,-4,-10]); K3
Knot represented by 6 crossings
sage: K3.dowker_notation()
[(5, 2), (4, 9), (1, 6), (7, 8), (10, 11), (12, 3)]
.. SEEALSO:: :meth:`~sage.knots.knot.Knot.dowker_notation`
REFERENCES:
- :wikipedia:`Dowker_notation`
- http://katlas.org/wiki/DT_(Dowker-Thistlethwaite)_Codes
"""
gauss = dowker_to_gauss(code)
orientations = recover_orientations(gauss)[3]
return Knot([[gauss], orientations])
def from_gauss_code(self, gauss):
"""
Build a knot from a signed Gauss code.
This makes some arbitrary choice of orientation.
INPUT:
- a signed Gauss code
OUTPUT:
- a knot
EXAMPLES::
sage: W = Knots()
sage: K1 = W.from_gauss_code([2, -1, 3, -2, 1, -3])
sage: K1.alexander_polynomial()
t^-1 - 1 + t
"""
orientations = recover_orientations(gauss)[3]
return Knot([[gauss], orientations])