# Knots#

AUTHORS:

• Miguel Angel Marco Buzunariz

class sage.knots.knot.Knot(data, check=True)[source]#

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.

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]))

>>> from sage.all import *
>>> B = BraidGroup(Integer(4))
>>> K = Knot(B([Integer(1),Integer(1),Integer(1),Integer(2),-Integer(1),Integer(2),-Integer(3),Integer(2),-Integer(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
sage: K.colored_jones_polynomial(2)  # long time
q^-1 - 2 + 4*q - 5*q^2 + 6*q^3 - 5*q^4 + 4*q^5 - 3*q^6 + q^7

>>> from sage.all import *
>>> K.alexander_polynomial()
-2*t^-2 + 8*t^-1 - 11 + 8*t - 2*t^2
>>> 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
>>> K.determinant()
31
>>> K.signature()
-2
>>> K.colored_jones_polynomial(Integer(2))  # long time
q^-1 - 2 + 4*q - 5*q^2 + 6*q^3 - 5*q^4 + 4*q^5 - 3*q^6 + q^7


REFERENCES:

arf_invariant()[source]#

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

>>> from sage.all import *
>>> B = BraidGroup(Integer(4))
>>> K = Knot(B([-Integer(1), Integer(2), Integer(1), Integer(2)]))
>>> K.arf_invariant()
0
>>> B = BraidGroup(Integer(8))
>>> K = Knot(B([-Integer(2), Integer(3), Integer(1), Integer(2), Integer(1), Integer(4)]))
>>> K.arf_invariant()
0
>>> K = Knot(B([Integer(1), Integer(2), Integer(1), Integer(2)]))
>>> K.arf_invariant()
1

colored_jones_polynomial(N, variab=None, try_inverse=True)[source]#

Return the colored Jones polynomial of the trace closure of the braid.

INPUT:

• N – integer; the number of colors

• variab – (default: $$q$$) the variable in the resulting Laurent polynomial

• try_inverse – boolean (default: True); if True, attempt a faster calculation by using the inverse of the braid

ALGORITHM:

The algorithm used is described in [HL2018] for the corresponding braid representation. We follow their notation, but work in a suitable free algebra over a Laurent polynomial ring in one variable to simplify bookkeeping.

EXAMPLES:

sage: W = Knots()
sage: K = W.from_dowker_code([-4,-6,-2])
sage: K.colored_jones_polynomial(2)
-q^-4 + q^-3 + q^-1
sage: K.colored_jones_polynomial(2, 't')
-t^-4 + t^-3 + t^-1
sage: R.<t> = LaurentPolynomialRing(ZZ)
sage: K.colored_jones_polynomial(2, -t)
-t^-4 - t^-3 - t^-1

sage: R.<t> = ZZ[]
sage: K.colored_jones_polynomial(2, t+1)
(t^3 + 3*t^2 + 4*t + 1)/(t^4 + 4*t^3 + 6*t^2 + 4*t + 1)

>>> from sage.all import *
>>> W = Knots()
>>> K = W.from_dowker_code([-Integer(4),-Integer(6),-Integer(2)])
>>> K.colored_jones_polynomial(Integer(2))
-q^-4 + q^-3 + q^-1
>>> K.colored_jones_polynomial(Integer(2), 't')
-t^-4 + t^-3 + t^-1
>>> R = LaurentPolynomialRing(ZZ, names=('t',)); (t,) = R._first_ngens(1)
>>> K.colored_jones_polynomial(Integer(2), -t)
-t^-4 - t^-3 - t^-1

>>> R = ZZ['t']; (t,) = R._first_ngens(1)
>>> K.colored_jones_polynomial(Integer(2), t+Integer(1))
(t^3 + 3*t^2 + 4*t + 1)/(t^4 + 4*t^3 + 6*t^2 + 4*t + 1)

connected_sum(other)[source]#

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

>>> from sage.all import *
>>> B = BraidGroup(Integer(2))
>>> trefoil = Knot(B([Integer(1),Integer(1),Integer(1)]))
>>> K = trefoil.connected_sum(trefoil); K
Knot represented by 6 crossings
>>> K.braid()
s0^3*s1^-1*s0^3*s1

sage: rev_trefoil = Knot(B([-1,-1,-1]))
sage: K2 = trefoil.connected_sum(rev_trefoil); K2
Knot represented by 6 crossings
sage: K2.braid()
s0^3*s1^-1*s0^-3*s1

>>> from sage.all import *
>>> rev_trefoil = Knot(B([-Integer(1),-Integer(1),-Integer(1)]))
>>> K2 = trefoil.connected_sum(rev_trefoil); K2
Knot represented by 6 crossings
>>> K2.braid()
s0^3*s1^-1*s0^-3*s1


Observe that both knots have according dowker_notation (showing that the constructing from DT-code may not be unique for non prime knots, see from_dowker_code()):

sage: K.dowker_notation()
[(4, 1), (2, 5), (6, 3), (10, 7), (8, 11), (12, 9)]
sage: K2.dowker_notation()
[(4, 1), (2, 5), (6, 3), (7, 10), (11, 8), (9, 12)]
sage: K.homfly_polynomial() == K2.homfly_polynomial()
False

>>> from sage.all import *
>>> K.dowker_notation()
[(4, 1), (2, 5), (6, 3), (10, 7), (8, 11), (12, 9)]
>>> K2.dowker_notation()
[(4, 1), (2, 5), (6, 3), (7, 10), (11, 8), (9, 12)]
>>> K.homfly_polynomial() == K2.homfly_polynomial()
False


REFERENCES:

dt_code()[source]#

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]

>>> from sage.all import *
>>> K = Knot([[Integer(1),Integer(5),Integer(2),Integer(4)],[Integer(5),Integer(3),Integer(6),Integer(2)],[Integer(3),Integer(1),Integer(4),Integer(6)]])
>>> K.dt_code()
[4, 6, 2]
>>> B = BraidGroup(Integer(4))
>>> K = Knot(B([Integer(1), Integer(2), Integer(1), Integer(2)]))
>>> K.dt_code()
[4, -6, 8, -2]
>>> K = Knot([[[Integer(1), -Integer(2), Integer(3), -Integer(4), Integer(5), -Integer(1), Integer(2), -Integer(3), Integer(4), -Integer(5)]],
...          [Integer(1), Integer(1), Integer(1), Integer(1), Integer(1)]])
>>> K.dt_code()
[6, 8, 10, 2, 4]

class sage.knots.knot.Knots[source]#

The set for all knots, as a monoid for the connected sum.

Element[source]#

alias of Knot

an_element()[source]#

Return the trefoil knot.

EXAMPLES:

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

>>> from sage.all import *
>>> Knots().an_element()
Knot represented by 3 crossings

from_dowker_code(code)[source]#

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

Warning

In general the Dowker-Thistlethwaite code does not describe a knot uniquely. It is not only insensitive on mirror images, but may also mix up non prime knots. For example [4, 6, 2, 10, 12, 8] describes the connected sum of two trefoil knots, as well as the connected sum of a trefoil with its mirror (see the corresponding example in the documentation of connected_sum()).

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)]

>>> from sage.all import *
>>> W = Knots()
>>> K1 = W.from_dowker_code([Integer(8),Integer(10),Integer(2),Integer(12),Integer(4),Integer(6)])
>>> K1.dowker_notation()
[(5, 2), (9, 4), (11, 6), (1, 8), (3, 10), (7, 12)]

>>> W.from_dowker_code([Integer(6),Integer(10),Integer(14),Integer(12),Integer(16),Integer(2),Integer(18),Integer(4),Integer(8)])
Knot represented by 9 crossings

>>> W.from_dowker_code([Integer(4),Integer(8),Integer(10),-Integer(14),Integer(2),-Integer(16),-Integer(18),-Integer(6),-Integer(12)])
Knot represented by 9 crossings

>>> K3 = W.from_dowker_code([Integer(6),-Integer(12),Integer(2),Integer(8),-Integer(4),-Integer(10)]); K3
Knot represented by 6 crossings
>>> K3.dowker_notation()
[(5, 2), (4, 9), (1, 6), (7, 8), (10, 11), (12, 3)]


dowker_notation()

REFERENCES:

from_gauss_code(gauss)[source]#

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 sage.all import *
>>> W = Knots()
>>> K1 = W.from_gauss_code([Integer(2), -Integer(1), Integer(3), -Integer(2), Integer(1), -Integer(3)])
>>> K1.alexander_polynomial()
t^-1 - 1 + t

from_table(n, k)[source]#

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

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

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

>>> K3 = Knots().from_table(Integer(10),Integer(56)); K3
Knot represented by 11 crossings
>>> 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

>>> K4 = Knots().from_table(Integer(10),Integer(100))
>>> K4.genus()
4


REFERENCES:

one()[source]#

Return the unit of the monoid.

This is the trivial knot.

EXAMPLES:

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

>>> from sage.all import *
>>> Knots().one()
Knot represented by 0 crossings