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 3-dimensional 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

Link

INPUT:

  • data – see Link for the allowable inputs
  • check – optional, default True. If True, 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]))
../../_images/knot-1.svg
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 and other.

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 and other.

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
../../_images/knot-2.svg
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
../../_images/knot-3.svg

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.

Element

alias of Knot

an_element()

Return the trefoil knot.

EXAMPLES:

sage: Knots().an_element()
Knot represented by 3 crossings
from_dowker_code(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)]

See also

dowker_notation()

REFERENCES:

from_gauss_code(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
from_table(n, k)

Return a knot from its index in the Rolfsen table.

INPUT:

  • n – the crossing number
  • k – a positive integer

OUTPUT:

the knot \(K_{n,k}\) in the Rolfsen table

EXAMPLES:

sage: K1 = Knots().from_table(6,3); K1
Knot represented by 6 crossings
sage: K1.alexander_polynomial()
t^-2 - 3*t^-1 + 5 - 3*t + t^2

sage: K2 = Knots().from_table(8,4); K2
Knot represented by 9 crossings
sage: K2.determinant()
19
sage: K2.signature()
2

sage: K3 = Knots().from_table(10,56); K3
Knot represented by 11 crossings
sage: K3.jones_polynomial()
t^10 - 3*t^9 + 6*t^8 - 9*t^7 + 10*t^6 - 11*t^5 + 10*t^4 - 7*t^3
+ 5*t^2 - 2*t + 1

sage: K4 = Knots().from_table(10,100)
sage: K4.genus()
4

REFERENCES:

one()

Return the unit of the monoid.

This is the trivial knot.

EXAMPLES:

sage: Knots().one()
Knot represented by 0 crossings