Knots¶
AUTHORS:
 Miguel Angel Marco Buzunariz
 Amit Jamadagni

class
sage.knots.knot.
Knot
(data, check=True)¶ Bases:
sage.knots.link.Link
,sage.structure.element.Element
A knot.
A knot is defined as embedding of the circle \(\mathbb{S}^1\) in the 3dimensional sphere \(\mathbb{S}^3\), considered up to ambient isotopy. They represent the physical idea of a knotted rope, but with the particularity that the rope is closed. That is, the ends of the rope are joined.
See also
INPUT:
data
– seeLink
for the allowable inputscheck
– optional, defaultTrue
. IfTrue
, make sure that the data define a knot, not a link
EXAMPLES:
We construct the knot \(8_{14}\) and compute some invariants:
sage: B = BraidGroup(4) sage: K = Knot(B([1,1,1,2,1,2,3,2,3]))
sage: K.alexander_polynomial() 2*t^2 + 8*t^1  11 + 8*t  2*t^2 sage: K.jones_polynomial() t^7  3*t^6 + 4*t^5  5*t^4 + 6*t^3  5*t^2 + 4*t + 1/t  2 sage: K.determinant() 31 sage: K.signature() 2
REFERENCES:

arf_invariant
()¶ Return the Arf invariant.
EXAMPLES:
sage: B = BraidGroup(4) sage: K = Knot(B([1, 2, 1, 2])) sage: K.arf_invariant() 0 sage: B = BraidGroup(8) sage: K = Knot(B([2, 3, 1, 2, 1, 4])) sage: K.arf_invariant() 0 sage: K = Knot(B([1, 2, 1, 2])) sage: K.arf_invariant() 1

connected_sum
(other)¶ Return the oriented connected sum of
self
andother
.Note
We give the knots an orientation based upon the braid representation.
INPUT:
other
– a knot
OUTPUT:
A knot equivalent to the connected sum of
self
andother
.EXAMPLES:
sage: B = BraidGroup(2) sage: trefoil = Knot(B([1,1,1])) sage: K = trefoil.connected_sum(trefoil); K Knot represented by 6 crossings sage: K.braid() s0^3*s1^1*s0^3*s1
sage: rev_trefoil = Knot(B([1,1,1])) sage: K = trefoil.connected_sum(rev_trefoil); K Knot represented by 6 crossings sage: K.braid() s0^3*s1^1*s0^3*s1
REFERENCES:

dt_code
()¶ Return the DT code of
self
.ALGORITHM:
The DT code is generated by the following way:
Start moving along the knot, as we encounter the crossings we start numbering them, so every crossing has two numbers assigned to it once we have traced the entire knot. Now we take the even number associated with every crossing.
The following sign convention is to be followed:
Take the even number with a negative sign if it is an overcrossing that we are encountering.
OUTPUT: DT code representation of the knot
EXAMPLES:
sage: K = Knot([[1,5,2,4],[5,3,6,2],[3,1,4,6]]) sage: K.dt_code() [4, 6, 2] sage: B = BraidGroup(4) sage: K = Knot(B([1, 2, 1, 2])) sage: K.dt_code() [4, 6, 8, 2] sage: K = Knot([[[1, 2, 3, 4, 5, 1, 2, 3, 4, 5]], [1, 1, 1, 1, 1]]) sage: K.dt_code() [6, 8, 10, 2, 4]

class
sage.knots.knot.
Knots
¶ Bases:
sage.misc.fast_methods.Singleton
,sage.structure.parent.Parent
The set for all knots, as a monoid for the connected sum.

an_element
()¶ Return the trefoil knot.
EXAMPLES:
sage: Knots().an_element() Knot represented by 3 crossings

one
()¶ Return the unit of the monoid.
This is the trivial knot.
EXAMPLES:
sage: Knots().one() Knot represented by 0 crossings
