Links¶

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.

A link is an embedding of one or more copies of \(\mathbb{S}^1\) in \(\mathbb{S}^3\), considered up to ambient isotopy. That is, a link represents the idea of one or more tied ropes. Every knot is a link, but not every link is a knot.

Generically, the projection of a link on \(\RR^2\) is a curve with crossings. The crossings are represented to show which strand goes over the other. This curve is called a planar diagram of the link. If we remove the crossings, the resulting connected components are segments. These segments are called the edges of the diagram.

REFERENCES:

See also

There are also tables of link and knot invariants at https://www.indiana.edu/~knotinfo/ and https://www.indiana.edu/~linkinfo/.

AUTHORS:

Miguel Angel Marco Buzunariz
Amit Jamadagni

class sage.knots.link.Link(data)¶

Bases: sage.structure.sage_object.SageObject

A link.

A link is an embedding of one or more copies of \(\mathbb{S}^1\) in \(\mathbb{S}^3\), considered up to ambient isotopy. That is, a link represents the idea of one or more tied ropes. Every knot is a link, but not every link is a knot.

A link can be created by using one of the conventions mentioned below:

Braid:

The closure of a braid is a link:

sage: B = BraidGroup(8)
sage: L = Link(B([-1, -1, -1, -2, 1, -2, 3, -2, 3]))
sage: L
Link with 1 component represented by 9 crossings
sage: L = Link(B([1, 2, 1, -2, -1]))
sage: L
Link with 2 components represented by 5 crossings

Note

The strands of the braid that have no crossings at all are removed.

Oriented Gauss Code:

Label the crossings from \(1\) to \(n\) (where \(n\) is the number of crossings) and start moving along the link. Trace every component of the link, by starting at a particular point on one component of the link and writing down each of the crossings that you encounter until returning to the starting point. The crossings are written with sign depending on whether we cross them as over or undercrossing. Each component is then represented as a list whose elements are the crossing numbers. A second list of \(+1\) and \(-1\)’s keeps track of the orientation of each crossing:
```
sage: L = Link([[[-1, 2, 3, -4, 5, -6, 7, 8, -2, -5, 6, 1, -8, -3, 4, -7]],
....:           [-1, -1, -1, -1, 1, 1, -1, 1]])
sage: L
Link with 1 component represented by 8 crossings
```
For links there may be more than one component and the input is as follows:
```
sage: L = Link([[[-1, 2], [-3, 4], [1, 3, -4, -2]], [-1, -1, 1, 1]])
sage: L
Link with 3 components represented by 4 crossings
```
Planar Diagram (PD) Code:

The diagram of the link is formed by segments that are adjacent to the crossings. Label each one of this segments with a positive number, and for each crossing, write down the four incident segments. The order of these segments is clockwise, starting with the incoming undercrossing.

There is no particular distinction between knots and links for this input.

EXAMPLES:

One of the representations of the trefoil knot:

sage: L = Link([[1, 5, 2, 4], [5, 3, 6, 2], [3, 1, 4, 6]])
sage: L
Link with 1 component represented by 3 crossings

One of the representations of the Hopf link:

sage: L = Link([[1, 4, 2, 3], [4, 1, 3, 2]])
sage: L
Link with 2 components represented by 2 crossings

We can construct links from the braid group:

sage: B = BraidGroup(4)
sage: L = Link(B([-1, -1, -1, -2, 1, -2, 3, -2]))
sage: L
Link with 2 components represented by 8 crossings

sage: L = Link(B([1, 2, 1, 3]))
sage: L
Link with 2 components represented by 4 crossings

We construct the “monster” unknot using a planar code, and then construct the oriented Gauss code and braid representation:

sage: L = Link([[3,1,2,4], [8,9,1,7], [5,6,7,3], [4,18,6,5],
....:           [17,19,8,18], [9,10,11,14], [10,12,13,11],
....:           [12,19,15,13], [20,16,14,15], [16,20,17,2]])
sage: L.oriented_gauss_code()
[[[1, -4, 3, -1, 10, -9, 6, -7, 8, 5, 4, -3, 2, -6, 7, -8, 9, -10, -5, -2]],
 [1, -1, 1, 1, 1, -1, -1, -1, -1, -1]]
sage: L.braid()
s0*s1^-3*s2^-1*s1*s3*s2^2*s1^-1*s0^-1*s2*s1^-1*s3^-1*s2*s1^-1

We construct the Ochiai unknot by using an oriented Gauss code:

sage: L = Link([[[1,-2,-3,-8,-12,13,-14,15,-7,-1,2,-4,10,11,-13,12,
....:             -11,-16,4,3,-5,6,-9,7,-15,14,16,-10,8,9,-6,5]],
....:           [-1,-1,1,1,1,1,-1,1,1,-1,1,-1,-1,-1,-1,-1]])
sage: L.pd_code()
[[10, 2, 11, 1], [2, 12, 3, 11], [3, 20, 4, 21], [12, 19, 13, 20],
 [21, 32, 22, 1], [31, 22, 32, 23], [9, 25, 10, 24], [4, 29, 5, 30],
 [23, 30, 24, 31], [28, 14, 29, 13], [17, 14, 18, 15], [5, 17, 6, 16],
 [15, 7, 16, 6], [7, 27, 8, 26], [25, 9, 26, 8], [18, 28, 19, 27]]

We construct the knot \(7_1\) and compute some invariants:

sage: B = BraidGroup(2)
sage: L = Link(B([1]*7))

sage: L.alexander_polynomial()
t^-3 - t^-2 + t^-1 - 1 + t - t^2 + t^3
sage: L.jones_polynomial()
-t^10 + t^9 - t^8 + t^7 - t^6 + t^5 + t^3
sage: L.determinant()
7
sage: L.signature()
-6

The links here have removed components in which no strand is used:

sage: B = BraidGroup(8)
sage: b = B([1])
sage: L = Link(b)
sage: b.components_in_closure()
7
sage: L.number_of_components()
1
sage: L.braid().components_in_closure()
1
sage: L.braid().parent()
Braid group on 2 strands

Warning

Equality of knots is done by comparing the corresponding braids, which may give false negatives.

Note

The behavior of removing unused strands from an element of a braid group may change without notice in the future. Do not rely on this feature.

Todo

Implement methods to creating new links from previously created links.

alexander_polynomial(var='t')¶

Return the Alexander polynomial of self.

INPUT:

var – (default: 't') the variable in the polynomial

EXAMPLES:

We begin by computing the Alexander polynomial for the figure-eight knot:

sage: B = BraidGroup(3)
sage: L = Link(B([1, -2, 1, -2]))
sage: L.alexander_polynomial()
-t^-1 + 3 - t

The “monster” unknot:

sage: L = Link([[3,1,2,4],[8,9,1,7],[5,6,7,3],[4,18,6,5],
....:           [17,19,8,18],[9,10,11,14],[10,12,13,11],
....:           [12,19,15,13],[20,16,14,15],[16,20,17,2]])
sage: L.alexander_polynomial()
1

Some additional examples:

sage: B = BraidGroup(2)
sage: L = Link(B([1]))
sage: L.alexander_polynomial()
1
sage: L = Link(B.one())
sage: L.alexander_polynomial()
1
sage: B = BraidGroup(3)
sage: L = Link(B([1, 2, 1, 2]))
sage: L.alexander_polynomial()
t^-1 - 1 + t

When the Seifert surface is disconnected, the Alexander polynomial is defined to be \(0\):

sage: B = BraidGroup(4)
sage: L = Link(B([1,3]))
sage: L.alexander_polynomial()
0

arcs(presentation='pd')¶

Return the arcs of self.

Arcs are the connected components of the planar diagram.

INPUT:

presentation – one of the following:
- 'pd' - the arcs are returned as lists of parts in the PD code
- 'gauss_code' - the arcs are returned as pieces of the Gauss code that start with a negative number, and end with the following negative one; of there exist a closed arc, it is returned as a list of positive numbers only

OUTPUT:

A list of lists representing the arcs based upon presentation.

EXAMPLES:

sage: K = Knot([[[1,-2,3,-1,2,-3]],[1,1,1]])
sage: K.arcs()
[[1, 2], [3, 4], [5, 6]]
sage: K.arcs(presentation='gauss_code')
[[-3, 1, -2], [-2, 3, -1], [-1, 2, -3]]

sage: L = Link([[1, 2, 3, 4], [3, 2, 1, 4]])
sage: L.arcs()
[[2, 4], [1], [3]]
sage: L.arcs(presentation='gauss_code')
[[-2, -1], [-1, -2], [2, 1]]
sage: L.gauss_code()
[[-1, -2], [2, 1]]

braid()¶

Return a braid representation of self.

OUTPUT: an element in the braid group

Warning

For the unknot with no crossings, this returns the identity of the braid group with 2 strands because this disregards strands with no crossings.

EXAMPLES:

sage: L = Link([[2, 3, 1, 4], [4, 1, 3, 2]])
sage: L.braid()
s^2
sage: L = Link([[[-1, 2, -3, 1, -2, 3]], [-1, -1, -1]])
sage: L.braid()
s^-3
sage: L = Link([[1,8,2,7], [8,4,9,5], [3,9,4,10], [10,1,7,6], [5,3,6,2]])
sage: L.braid()
(s0*s1^-1)^2*s1^-1

colorings(n)¶

Return the n-colorings of self.

INPUT:

n – the number of colors to consider

If n is not a prime number, it is replaced by the smallest prime number that is larger than n.

OUTPUT:

a list with the colorings. Each coloring is represented as a dictionary that maps a tuple of the edges forming each arc (as in the PD code) to the index of the corresponding color.

EXAMPLES:

sage: K = Link([[[1, -2, 3, -1, 2, -3]], [1, 1, 1]])
sage: K.colorings(3)
[{(1, 2): 0, (3, 4): 1, (5, 6): 2},
 {(1, 2): 0, (3, 4): 2, (5, 6): 1},
 {(1, 2): 1, (3, 4): 0, (5, 6): 2},
 {(1, 2): 1, (3, 4): 2, (5, 6): 0},
 {(1, 2): 2, (3, 4): 0, (5, 6): 1},
 {(1, 2): 2, (3, 4): 1, (5, 6): 0}]
sage: K.pd_code()
[[4, 1, 5, 2], [2, 5, 3, 6], [6, 3, 1, 4]]
sage: K.arcs('pd')
[[1, 2], [3, 4], [5, 6]]

REFERENCES:

Wikipedia article Fox_n-coloring
Chapter 3 of [Liv1993]

See also

is_colorable()

determinant()¶

Return the determinant of self.

EXAMPLES:

sage: B = BraidGroup(4)
sage: L = Link(B([-1, 2, 1, 2]))
sage: L.determinant()
1
sage: B = BraidGroup(8)
sage: L = Link(B([2, 4, 2, 3, 1, 2]))
sage: L.determinant()
3
sage: L = Link(B([1]*16 + [2,1,2,1,2,2,2,2,2,2,2,1,2,1,2,-1,2,-2]))
sage: L.determinant()
65

dowker_notation()¶

Return the Dowker notation of self.

Similar to the PD code we number the components, so every crossing is represented by four numbers. We focus on the incoming entities of the under and the overcrossing. It is the pair of incoming undercrossing and the incoming overcrossing. This information at every crossing gives the Dowker notation.

OUTPUT:

A list containing the pair of incoming under cross and the incoming over cross.

EXAMPLES:

sage: L = Link([[[-1, 2, -3, 4, 5, 1, -2, 6, 7, 3, -4, -7, -6,-5]], [-1, -1, -1, -1, 1, -1, 1]])
sage: L.dowker_notation()
[(1, 6), (7, 2), (3, 10), (11, 4), (14, 5), (13, 8), (12, 9)]
sage: B = BraidGroup(4)
sage: L = Link(B([1, 2, 1, 2]))
sage: L.dowker_notation()
[(2, 1), (3, 5), (6, 4), (7, 9)]
sage: L = Link([[1, 4, 2, 3], [4, 1, 3, 2]])
sage: L.dowker_notation()
[(1, 3), (4, 2)]

fundamental_group(presentation='wirtinger')¶

Return the fundamental group of the complement of self.

INPUT:

presentation – string; one of the following:
- 'wirtinger' - (default) the Wirtinger presentation (see Wikipedia article Link_group)
- 'braid' - the presentation is given by the braid action on the free group (see chapter 2 of [Bir1975])

OUTPUT:

a finitely presented group

EXAMPLES:

sage: L = Link([[1, 2, 3, 4], [3, 2, 1, 4]])
sage: L.fundamental_group()
Finitely presented group < x0, x1, x2 | x1*x0^-1*x2^-1*x0, x2*x0*x1^-1*x0^-1 >
sage: L.fundamental_group('braid')
Finitely presented group < x0, x1 | 1, 1 >

We can see, for instance, that the two presentations of the group of the figure eight knot correspond to isomorphic groups:

sage: K8 = Knot([[[1, -2, 4, -3, 2, -1, 3, -4]], [1, 1, -1, -1]])
sage: GA = K8.fundamental_group()
sage: GA
Finitely presented group < x0, x1, x2, x3 |
 x2*x0*x3^-1*x0^-1, x0*x2*x1^-1*x2^-1,
 x1*x3^-1*x2^-1*x3, x3*x1^-1*x0^-1*x1 >
sage: GB = K8.fundamental_group(presentation='braid')
sage: GB
Finitely presented group < x0, x1, x2 | x1*x2^-1*x1^-1*x0*x1*x2*x1*x2^-1*x1^-1*x0^-1*x1*x2*x1^-1*x0^-1, x1*x2^-1*x1^-1*x0*x1*x2*x1^-1*x2^-1*x1^-1*x0^-1*x1*x2*x1^-1*x0*x1*x2*x1*x2^-1*x1^-1*x0^-1*x1*x2*x1^-2, x1*x2^-1*x1^-1*x0*x1*x2*x1^-1*x2^-1 >
sage: GA.simplified()
Finitely presented group < x0, x1 |
 x1^-1*x0*x1*x0^-1*x1*x0*x1^-1*x0^-1*x1*x0^-1 >
sage: GB.simplified()
Finitely presented group < x0, x2 |
 x2^-1*x0*x2^-1*x0^-1*x2*x0*x2^-1*x0*x2*x0^-1 >

gauss_code()¶

Return the Gauss code of self.

The Gauss code is generated by the following procedure:

Number the crossings from \(1\) to \(n\).
Select a point on the knot and start moving along the component.
At each crossing, take the number of the crossing, along with sign, which is \(-\) if it is an undercrossing and \(+\) if it is a overcrossing.

EXAMPLES:

sage: L = Link([[1, 4, 2, 3], [4, 1, 3, 2]])
sage: L.gauss_code()
[[-1, 2], [1, -2]]
sage: B = BraidGroup(8)
sage: L = Link(B([1, -2, 1, -2, -2]))
sage: L.gauss_code()
[[-1, 3, -4, 5], [1, -2, 4, -5, 2, -3]]
sage: L = Link([[[-1, 2], [-3, 4], [1, 3, -4, -2]], [-1, -1, 1, 1]])
sage: L.gauss_code()
[[-1, 2], [-3, 4], [1, 3, -4, -2]]

genus()¶

Return the genus of self.

EXAMPLES:

sage: B = BraidGroup(4)
sage: L = Link(B([-1, 3, 1, 3]))
sage: L.genus()
0
sage: L = Link(B([1,3]))
sage: L.genus()
0
sage: B = BraidGroup(8)
sage: L = Link(B([-2, 4, 1, 6, 1, 4]))
sage: L.genus()
0
sage: L = Link(B([1, 2, 1, 2]))
sage: L.genus()
1

homfly_polynomial(var1='L', var2='M', normalization='lm')¶

Return the HOMFLY polynomial of self.

The HOMFLY polynomial \(P(K)\) of a link \(K\) is a Laurent polynomial in two variables defined using skein relations and for the unknot \(U\), we have \(P(U) = 1\).

INPUT:

var1 – (default: 'L') the first variable
var2 – (default: 'M') the second variable
normalization – (default: lm) the system of coordinates and can be one of the following:
- 'lm' – corresponding to the Skein relation \(L\cdot P(K _+) + L^{-1}\cdot P(K _-) + M\cdot P(K _0) = 0\)
- 'az' – corresponding to the Skein relation \(a\cdot P(K _+) - a^{-1}\cdot P(K _-) = z \cdot P(K _0)\)
where \(P(K _+)\), \(P(K _-)\) and \(P(K _0)\) represent the HOMFLY polynomials of three links that vary only in one crossing; that is the positive, negative, or smoothed links respectively

OUTPUT:

A Laurent polynomial over the integers.

Note

Use the 'az' normalization to agree with the data in [KnotAtlas] and http://www.indiana.edu/~knotinfo/.

EXAMPLES:

We give some examples:

sage: g = BraidGroup(2).gen(0)
sage: K = Knot(g^5)
sage: K.homfly_polynomial()
L^-4*M^4 - 4*L^-4*M^2 + 3*L^-4 - L^-6*M^2 + 2*L^-6

The Hopf link:

sage: L = Link([[1,3,2,4],[4,2,3,1]])
sage: L.homfly_polynomial('x', 'y')
-x^-1*y + x^-1*y^-1 + x^-3*y^-1

Another version of the Hopf link where the orientation has been changed. Therefore we substitute \(x \mapsto L^{-1}\) and \(y \mapsto M\):

sage: L = Link([[1,4,2,3], [4,1,3,2]])
sage: L.homfly_polynomial()
L^3*M^-1 - L*M + L*M^-1
sage: L = Link([[1,4,2,3], [4,1,3,2]])
sage: L.homfly_polynomial('a', 'z', 'az')
a^3*z^-1 - a*z - a*z^-1

The figure-eight knot:

sage: L = Link([[2,1,4,5], [5,6,7,3], [6,4,1,9], [9,2,3,7]])
sage: L.homfly_polynomial()
-L^2 + M^2 - 1 - L^-2
sage: L.homfly_polynomial('a', 'z', 'az')
a^2 - z^2 - 1 + a^-2

The “monster” unknot:

sage: L = Link([[3,1,2,4], [8,9,1,7], [5,6,7,3], [4,18,6,5],
....:           [17,19,8,18], [9,10,11,14], [10,12,13,11],
....:           [12,19,15,13], [20,16,14,15], [16,20,17,2]])
sage: L.homfly_polynomial()
1

The knot \(9_6\):

sage: B = BraidGroup(3)
sage: K = Knot(B([-1,-1,-1,-1,-1,-1,-2,1,-2,-2]))
sage: K.homfly_polynomial()
L^10*M^4 - L^8*M^6 - 3*L^10*M^2 + 4*L^8*M^4 + L^6*M^6 + L^10
 - 3*L^8*M^2 - 5*L^6*M^4 - L^8 + 7*L^6*M^2 - 3*L^6
sage: K.homfly_polynomial('a', 'z', normalization='az')
-a^10*z^4 + a^8*z^6 - 3*a^10*z^2 + 4*a^8*z^4 + a^6*z^6 - a^10
 + 3*a^8*z^2 + 5*a^6*z^4 - a^8 + 7*a^6*z^2 + 3*a^6

REFERENCES:

is_alternating()¶

Return True if the given knot diagram is alternating else returns False.

Alternating diagram implies every overcross is followed by an undercross or the vice-versa.

We look at the Gauss code if the sign is alternating, True is returned else the knot is not alternating False is returned.

Warning

This does not check if a knot admits an alternating diagram or not. Thus, this term is used differently than in some of the literature, such as in Hoste-Thistlethwaite table.

Note

Links with more than one component are considered to not be alternating (knots) even when such a diagram exists.

EXAMPLES:

sage: B = BraidGroup(4)
sage: L = Link(B([-1, -1, -1, -1]))
sage: L.is_alternating()
False
sage: L = Link(B([1, -2, -1, 2]))
sage: L.is_alternating()
False
sage: L = Link(B([-1, 3, 1, 3, 2]))
sage: L.is_alternating()
False
sage: L = Link(B([1]*16 + [2,1,2,1,2,2,2,2,2,2,2,1,2,1,2,-1,2,-2]))
sage: L.is_alternating()
False
sage: L = Link(B([-1,2,-1,2]))
sage: L.is_alternating()
True

We give the \(5_2\) knot with an alternating diagram and a non-alternating diagram:

sage: K5_2 = Link([[1, 4, 2, 5], [3, 8, 4, 9], [5, 10, 6, 1],
....:              [7, 2, 8, 3], [9, 6, 10, 7]])
sage: K5_2.is_alternating()
True

sage: K5_2b = Link(K5_2.braid())
sage: K5_2b.is_alternating()
False

is_colorable(n)¶

Return whether the link is n-colorable.

A link is n-colorable if its arcs can be painted with n colours, labeled from 0 to n - 1, in such a way that at any crossing, the average of the indices of the undercrossings equals twice the index of the overcrossing.

INPUT:

n – the number of colors to consider

If n is not a prime number, it is replaced by the smallest prime number that is larger than n.

EXAMPLES:

We show that the trefoil knot is 3-colorable:

sage: K = Link([[[1, -2, 3, -1, 2, -3]], [1, 1, 1]])
sage: K.is_colorable(3)
True

But the figure eight knot is not:

sage: K8 = Link([[[1, -2, 4, -3, 2, -1, 3, -4]], [1, 1, -1, -1]])
sage: K8.is_colorable(3)
False

REFERENCES:

Wikipedia article Fox_n-coloring
Chapter 3 of [Liv1993]

See also

colorings()

is_knot()¶

Return True if self is a knot.

Every knot is a link but the converse is not true.

EXAMPLES:

sage: B = BraidGroup(4)
sage: L = Link(B([1, 3, 1, -3]))
sage: L.is_knot()
False
sage: B = BraidGroup(8)
sage: L = Link(B([1, 2, 3, 4, 5, 6]))
sage: L.is_knot()
True

jones_polynomial(variab=None, skein_normalization=False, algorithm='jonesrep')¶

Return the Jones polynomial of self.

The normalization is so that the unknot has Jones polynomial \(1\). If skein_normalization is True, the variable of the result is replaced by a itself to the power of \(4\), so that the result agrees with the conventions of [Lic1997] (which in particular differs slightly from the conventions used otherwise in this class), had one used the conventional Kauffman bracket variable notation directly.

If variab is None return a polynomial in the variable \(A\) or \(t\), depending on the value skein_normalization. In particular, if skein_normalization is False, return the result in terms of the variable \(t\), also used in [Lic1997].

ALGORITHM:

The calculation goes through one of two possible algorithms, depending on the value of algorithm. Possible values are 'jonesrep' which uses the Jones representation of a braid representation of self to compute the polynomial of the trace closure of the braid, and statesum which recursively computes the Kauffman bracket of self. Depending on how the link is given, there might be significant time gains in using one over the other. When the trace closure of the braid is self, the algorithms give the same result.

INPUT:

variab – variable (default: None); the variable in the resulting polynomial; if unspecified, use either a default variable in \(\ZZ[A,A^{-1}]\) or the variable \(t\) in the symbolic ring
skein_normalization – boolean (default: False); determines the variable of the resulting polynomial
algorithm – string (default: 'jonesrep'); algorithm to use and can be one of the following:
- 'jonesrep' - use the Jones representation of the braid representation
- 'statesum' - recursively computes the Kauffman bracket

OUTPUT:

If skein_normalization if False, this returns an element in the symbolic ring as the Jones polynomial of the link might have fractional powers when the link is not a knot. Otherwise the result is a Laurent polynomial in variab.

EXAMPLES:

The unknot:

sage: B = BraidGroup(9)
sage: b = B([1, 2, 3, 4, 5, 6, 7, 8])
sage: Link(b).jones_polynomial()
1

The “monster” unknot:

sage: L = Link([[3,1,2,4],[8,9,1,7],[5,6,7,3],[4,18,6,5],
....:           [17,19,8,18],[9,10,11,14],[10,12,13,11],
....:           [12,19,15,13],[20,16,14,15],[16,20,17,2]])
sage: L.jones_polynomial()
1

The Ochiai unknot:

sage: L = Link([[[1,-2,-3,-8,-12,13,-14,15,-7,-1,2,-4,10,11,-13,12,
....:             -11,-16,4,3,-5,6,-9,7,-15,14,16,-10,8,9,-6,5]],
....:           [-1,-1,1,1,1,1,-1,1,1,-1,1,-1,-1,-1,-1,-1]])
sage: L.jones_polynomial()  # long time
1

Two unlinked unknots:

sage: B = BraidGroup(4)
sage: b = B([1, 3])
sage: Link(b).jones_polynomial()
-sqrt(t) - 1/sqrt(t)

The Hopf link:

sage: B = BraidGroup(2)
sage: b = B([-1,-1])
sage: Link(b).jones_polynomial()
-1/sqrt(t) - 1/t^(5/2)

Different representations of the trefoil and one of its mirror:

sage: B = BraidGroup(2)
sage: b = B([-1, -1, -1])
sage: Link(b).jones_polynomial(skein_normalization=True)
-A^-16 + A^-12 + A^-4
sage: Link(b).jones_polynomial()
1/t + 1/t^3 - 1/t^4
sage: B = BraidGroup(3)
sage: b = B([-1, -2, -1, -2])
sage: Link(b).jones_polynomial(skein_normalization=True)
-A^-16 + A^-12 + A^-4
sage: R.<x> = LaurentPolynomialRing(GF(2))
sage: Link(b).jones_polynomial(skein_normalization=True, variab=x)
x^-16 + x^-12 + x^-4
sage: B = BraidGroup(3)
sage: b = B([1, 2, 1, 2])
sage: Link(b).jones_polynomial(skein_normalization=True)
A^4 + A^12 - A^16

\(K11n42\) (the mirror of the “Kinoshita-Terasaka” knot) and \(K11n34\) (the mirror of the “Conway” knot) in [KnotAtlas]:

sage: B = BraidGroup(4)
sage: K11n42 = Link(B([1, -2, 3, -2, 3, -2, -2, -1, 2, -3, -3, 2, 2]))
sage: K11n34 = Link(B([1, 1, 2, -3, 2, -3, 1, -2, -2, -3, -3]))
sage: bool(K11n42.jones_polynomial() == K11n34.jones_polynomial())
True

The two algorithms for computation give the same result when the trace closure of the braid representation is the link itself:

sage: L = Link([[[-1, 2, -3, 4, 5, 1, -2, 6, 7, 3, -4, -7, -6, -5]],
....:           [-1, -1, -1, -1, 1, -1, 1]])
sage: jonesrep = L.jones_polynomial(algorithm='jonesrep')
sage: statesum = L.jones_polynomial(algorithm='statesum')
sage: bool(jonesrep == statesum)
True

When we have thrown away unknots so that the trace closure of the braid is not necessarily the link itself, this is only true up to a power of the Jones polynomial of the unknot:

sage: B = BraidGroup(3)
sage: b = B([1])
sage: L = Link(b)
sage: b.components_in_closure()
2
sage: L.number_of_components()
1
sage: b.jones_polynomial()
-sqrt(t) - 1/sqrt(t)
sage: L.jones_polynomial()
1
sage: L.jones_polynomial(algorithm='statesum')
1

khovanov_homology(ring=Integer Ring, height=None, degree=None)¶

Return the Khovanov homology of the link.

INPUT:

ring – (default: ZZ) the coefficient ring
height – the height of the homology to compute, if not specified, all the heights are computed
degree – the degree of the homology to compute, if not specified, all the degrees are computed

OUTPUT:

The Khovanov homology of the Link. It is given as a dictionary whose keys are the different heights. For each height, the homology is given as another dictionary whose keys are the degrees.

EXAMPLES:

sage: K = Link([[[1, -2, 3, -1, 2, -3]],[-1, -1, -1]])
sage: K.khovanov_homology()
{-9: {-3: Z},
 -7: {-3: 0, -2: C2},
 -5: {-3: 0, -2: Z, -1: 0, 0: 0},
 -3: {-3: 0, -2: 0, -1: 0, 0: Z},
 -1: {0: Z}}

The figure eight knot:

sage: L = Link([[1, 6, 2, 7], [5, 2, 6, 3], [3, 1, 4, 8], [7, 5, 8, 4]])
sage: L.khovanov_homology(height=-1)
{-1: {-2: 0, -1: Z, 0: Z, 1: 0, 2: 0}}

The Hopf link:

sage: B = BraidGroup(2)
sage: b = B([1, 1])
sage: K = Link(b)
sage: K.khovanov_homology(degree = 2)
{2: {2: 0}, 4: {2: Z}, 6: {2: Z}}

mirror_image()¶

Return the mirror image of self.

EXAMPLES:

sage: g = BraidGroup(2).gen(0)
sage: K = Link(g^3)
sage: K2 = K.mirror_image(); K2
Link with 1 component represented by 3 crossings
sage: K2.braid()
s^-3

sage: K = Knot([[[1, -2, 3, -1, 2, -3]], [1, 1, 1]])
sage: K2 = K.mirror_image(); K2
Knot represented by 3 crossings
sage: K.pd_code()
[[4, 1, 5, 2], [2, 5, 3, 6], [6, 3, 1, 4]]
sage: K2.pd_code()
[[4, 2, 5, 1], [2, 6, 3, 5], [6, 4, 1, 3]]

number_of_components()¶

Return the number of connected components of self.

OUTPUT: number of connected components

EXAMPLES:

sage: B = BraidGroup(4)
sage: L = Link(B([-1, 3, 1, 3]))
sage: L.number_of_components()
4
sage: B = BraidGroup(8)
sage: L = Link(B([-2, 4, 1, 6, 1, 4]))
sage: L.number_of_components()
5
sage: L = Link(B([1, 2, 1, 2]))
sage: L.number_of_components()
1
sage: L = Link(B.one())
sage: L.number_of_components()
1

omega_signature(omega)¶

Compute the \(\omega\)-signature of self.

INPUT:

\(\omega\) – a complex number of modulus 1. This is assumed to be coercible to QQbar.

This is defined as the signature of the Hermitian matrix

\[(1 - \omega) V + (1 - \omega^{-1}) V^{t},\]

where \(V\) is the Seifert matrix, as explained on page 122 of [Liv1993].

According to [Con2018], this is also known as the Levine-Tristram signature, the equivariant signature or the Tristram-Levine signature.

See also

signature(), seifert_matrix()

EXAMPLES:

sage: B = BraidGroup(4)
sage: K = Knot(B([1,1,1,2,-1,2,-3,2,-3]))
sage: omega = QQbar.zeta(3)
sage: K.omega_signature(omega)
-2

orientation()¶

Return the orientation of the crossings of the link diagram of self.

EXAMPLES:

sage: L = Link([[1, 4, 5, 2], [3, 5, 6, 7], [4, 8, 9, 6], [7, 9, 10, 11], [8, 1, 13, 10], [11, 13, 2, 3]])
sage: L.orientation()
[-1, 1, -1, 1, -1, 1]
sage: L = Link([[1, 7, 2, 6], [7, 3, 8, 2], [3, 11, 4, 10], [11, 5, 12, 4], [14, 5, 1, 6], [13, 9, 14, 8], [12, 9, 13, 10]])
sage: L.orientation()
[-1, -1, -1, -1, 1, -1, 1]
sage: L = Link([[1, 2, 3, 3], [2, 4, 5, 5], [4, 1, 7, 7]])
sage: L.orientation()
[-1, -1, -1]

oriented_gauss_code()¶

Return the oriented Gauss code of self.

The oriented Gauss code has two parts:

the Gauss code
the orientation of each crossing

The following orientation was taken into consideration for construction of knots:

From the outgoing of the overcrossing if we move in the clockwise direction to reach the outgoing of the undercrossing then we label that crossing as \(-1\).

From the outgoing of the overcrossing if we move in the anticlockwise direction to reach the outgoing of the undercrossing then we label that crossing as \(+1\).

One more consideration we take in while constructing the orientation is the order of the orientation is same as the ordering of the crossings in the Gauss code.

Note

Convention: under is denoted by \(-1\), and over by \(+1\) in the crossing info.

EXAMPLES:

sage: L = Link([[1, 11, 2, 10], [6, 2, 7, 3], [3, 12, 4, 9], [9, 5, 10, 6], [8, 1, 5, 4], [11, 8, 12, 7]])
sage: L.oriented_gauss_code()
[[[-1, 2, -3, 5], [4, -2, 6, -5], [-4, 1, -6, 3]], [-1, 1, 1, 1, -1, -1]]
sage: L = Link([[1, 4, 2, 3], [6, 1, 3, 2], [7, 4, 8, 5], [5, 8, 6, 7]])
sage: L.oriented_gauss_code()
[[[-1, 2], [-3, 4], [1, 3, -4, -2]], [-1, -1, 1, 1]]
sage: B = BraidGroup(8)
sage: b = B([1, 1, 1, 1, 1])
sage: L = Link(b)
sage: L.oriented_gauss_code()
[[[1, -2, 3, -4, 5, -1, 2, -3, 4, -5]], [1, 1, 1, 1, 1]]

pd_code()¶

Return the planar diagram code of self.

The planar diagram is returned in the following format.

We construct the crossing by starting with the entering component of the undercrossing, move in the clockwise direction and then generate the list. If the crossing is given by \([a, b, c, d]\), then we interpret this information as:

\(a\) is the entering component of the undercrossing;
\(b, d\) are the components of the overcrossing;
\(c\) is the leaving component of the undercrossing.

EXAMPLES:

sage: L = Link([[[1, -2, 3, -4, 2, -1, 4, -3]], [1, 1, -1, -1]])
sage: L.pd_code()
[[6, 1, 7, 2], [2, 5, 3, 6], [8, 4, 1, 3], [4, 8, 5, 7]]
sage: B = BraidGroup(2)
sage: b = B([1, 1, 1, 1, 1])
sage: L = Link(b)
sage: L.pd_code()
[[2, 1, 3, 4], [4, 3, 5, 6], [6, 5, 7, 8], [8, 7, 9, 10], [10, 9, 1, 2]]
sage: L = Link([[[2, -1], [1, -2]], [1, 1]])
sage: L.pd_code()
[[2, 3, 1, 4], [4, 1, 3, 2]]
sage: L = Link([[1, 2, 3, 3], [2, 4, 5, 5], [4, 1, 7, 7]])
sage: L.pd_code()
[[1, 2, 3, 3], [2, 4, 5, 5], [4, 1, 7, 7]]

plot(gap=0.1, component_gap=0.5, solver=None, color='blue', **kwargs)¶

Plot self.

INPUT:

gap – (default: 0.1) the size of the blank gap left for the crossings
component_gap – (default: 0.5) the gap between isolated components
solver – the linear solver to use, see MixedIntegerLinearProgram.
color – (default: ‘blue’) a color or a coloring (as returned by colorings().

The usual keywords for plots can be used here too.

EXAMPLES:

We construct the simplest version of the unknot:

sage: L = Link([[2, 1, 1, 2]])
sage: L.plot()
Graphics object consisting of ... graphics primitives

We construct a more interesting example of the unknot:

sage: L = Link([[2, 1, 4, 5], [3, 5, 6, 7], [4, 1, 9, 6], [9, 2, 3, 7]])
sage: L.plot()
Graphics object consisting of ... graphics primitives

The “monster” unknot:

sage: L = Link([[3,1,2,4],[8,9,1,7],[5,6,7,3],[4,18,6,5],
....:           [17,19,8,18],[9,10,11,14],[10,12,13,11],
....:           [12,19,15,13],[20,16,14,15],[16,20,17,2]])
sage: L.plot()
Graphics object consisting of ... graphics primitives

The Ochiai unknot:

sage: L = Link([[[1,-2,-3,-8,-12,13,-14,15,-7,-1,2,-4,10,11,-13,12,
....:             -11,-16,4,3,-5,6,-9,7,-15,14,16,-10,8,9,-6,5]],
....:           [-1,-1,1,1,1,1,-1,1,1,-1,1,-1,-1,-1,-1,-1]])
sage: L.plot()
Graphics object consisting of ... graphics primitives

One of the representations of the trefoil knot:

sage: L = Link([[1, 5, 2, 4], [5, 3, 6, 2], [3, 1, 4, 6]])
sage: L.plot()
Graphics object consisting of 14 graphics primitives

The figure-eight knot:

sage: L = Link([[2, 1, 4, 5], [5, 6, 7, 3], [6, 4, 1, 9], [9, 2, 3, 7]])
sage: L.plot()
Graphics object consisting of ... graphics primitives

The knot \(K11n121\) in [KnotAtlas]:

sage: L = Link([[4,2,5,1], [10,3,11,4], [5,16,6,17], [7,12,8,13],
....:           [18,9,19,10], [2,11,3,12], [13,20,14,21], [15,6,16,7],
....:           [22,18,1,17], [8,19,9,20], [21,14,22,15]])
sage: L.plot()
Graphics object consisting of ... graphics primitives

One of the representations of the Hopf link:

sage: L = Link([[1, 4, 2, 3], [4, 1, 3, 2]])
sage: L.plot()
Graphics object consisting of ... graphics primitives

Plotting links with multiple isolated components:

sage: L = Link([[[-1, 2, -3, 1, -2, 3], [4, -5, 6, -4, 5, -6]], [1, 1, 1, 1, 1, 1]])
sage: L.plot()
Graphics object consisting of ... graphics primitives

If a coloring is passed, the different arcs are plotted with the corresponding colors:

sage: B = BraidGroup(4)
sage: b = B([1,2,3,1,2,-1,-3,2,3])
sage: L = Link(b)
sage: L.plot(color=L.colorings(3)[0])
Graphics object consisting of ... graphics primitives

regions()¶

Return the regions from the link diagram of self.

Regions are obtained always turning left at each crossing.

Then the regions are represented as a list with the segments that form its boundary, with a sign depending on the orientation of the segment as part of the boundary.

EXAMPLES:

sage: L = Link([[[-1, +2, -3, 4, +5, +1, -2, +6, +7, 3, -4, -7, -6,-5]],[-1, -1, -1, -1, 1, -1, 1]])
sage: L.regions()
[[14, -5, 12, -9], [13, 9], [11, 5, 1, 7, 3], [10, -3, 8, -13], [6, -1], [4, -11], [2, -7], [-2, -6, -14, -8], [-4, -10, -12]]
sage: L = Link([[[1, -2, 3, -4, 2, -1, 4, -3]],[1, 1, -1, -1]])
sage: L.regions()
[[8, 4], [7, -4, 1], [6, -1, -3], [5, 3, -8], [2, -5, -7], [-2, -6]]
sage: L = Link([[[-1, +2, 3, -4, 5, -6, 7, 8, -2, -5, +6, +1, -8, -3, 4, -7]],[-1, -1, -1, -1, 1, 1, -1, 1]])
sage: L.regions()
[[16, 8, 14, 4], [15, -4], [13, -8, 1], [12, -1, -7], [11, 7, -16, 5], [10, -5, -15, -3], [9, 3, -14], [6, -11], [2, -9, -13], [-2, -12, -6, -10]]
sage: B = BraidGroup(2)
sage: L = Link(B([-1, -1, -1]))
sage: L.regions()
[[6, -5], [5, 1, 3], [4, -3], [2, -1], [-2, -6, -4]]
sage: L = Link([[[1, -2, 3, -4], [-1, 5, -3, 2, -5, 4]], [-1, 1, 1, -1, -1]])
sage: L.regions()
[[10, -4, -7], [9, 7, -3], [8, 3], [6, -9, -2], [5, 2, -8, 4], [1, -5], [-1, -10, -6]]
sage: L = Link([[1, 2, 3, 3], [2, 5, 4, 4], [5, 7, 6, 6], [7, 1, 8, 8]])
sage: L.regions()
[[-3], [-4], [-6], [-8], [7, 1, 2, 5], [-1, 8, -7, 6, -5, 4, -2, 3]]

Note

The link diagram is assumed to have only one completely isolated component. This is because otherwise some regions would have disconnected boundary.

seifert_circles()¶

Return the Seifert circles from the link diagram of self.

Seifert circles are the circles obtained by smoothing all crossings respecting the orientation of the segments.

Each Seifert circle is represented as a list of the segments that form it.

EXAMPLES:

sage: L = Link([[[1, -2, 3, -4, 2, -1, 4, -3]], [1, 1, -1, -1]])
sage: L.seifert_circles()
[[1, 7, 5, 3], [2, 6], [4, 8]]
sage: L = Link([[[-1, 2, 3, -4, 5, -6, 7, 8, -2, -5, 6, 1, -8, -3, 4, -7]], [-1, -1, -1, -1, 1, 1, -1, 1]])
sage: L.seifert_circles()
[[1, 13, 9, 3, 15, 5, 11, 7], [2, 10, 6, 12], [4, 16, 8, 14]]
sage: L = Link([[[-1, 2, -3, 4, 5, 1, -2, 6, 7, 3, -4, -7, -6, -5]], [-1, -1, -1, -1, 1, -1, 1]])
sage: L.seifert_circles()
[[1, 7, 3, 11, 5], [2, 8, 14, 6], [4, 12, 10], [9, 13]]
sage: L = Link([[1, 7, 2, 6], [7, 3, 8, 2], [3, 11, 4, 10], [11, 5, 12, 4], [14, 5, 1, 6], [13, 9, 14, 8], [12, 9, 13, 10]])
sage: L.seifert_circles()
[[1, 7, 3, 11, 5], [2, 8, 14, 6], [4, 12, 10], [9, 13]]
sage: L = Link([[[-1, 2, -3, 5], [4, -2, 6, -5], [-4, 1, -6, 3]], [-1, 1, 1, 1, -1, -1]])
sage: L.seifert_circles()
[[1, 11, 8], [2, 7, 12, 4, 5, 10], [3, 9, 6]]
sage: B = BraidGroup(2)
sage: L = Link(B([1, 1, 1]))
sage: L.seifert_circles()
[[1, 3, 5], [2, 4, 6]]

seifert_matrix()¶

Return the Seifert matrix associated with self.

ALGORITHM:

This is the algorithm presented in Section 3.3 of [Col2013].

OUTPUT:

The intersection matrix of a (not necessarily minimal) Seifert surface.

EXAMPLES:

sage: B = BraidGroup(4)
sage: L = Link(B([-1, 3, 1, 3]))
sage: L.seifert_matrix()
[ 0  0]
[ 0 -1]
sage: B = BraidGroup(8)
sage: L = Link(B([-1, 3, 1, 5, 1, 7, 1, 6]))
sage: L.seifert_matrix()
[ 0  0  0]
[ 1 -1  0]
[ 0  1 -1]
sage: L = Link(B([-2, 4, 1, 6, 1, 4]))
sage: L.seifert_matrix()
[-1  0]
[ 0 -1]

signature()¶

Return the signature of self.

This is defined as the signature of the symmetric matrix

\[V + V^{t},\]

where \(V\) is the Seifert matrix.

See also

omega_signature(), seifert_matrix()

EXAMPLES:

sage: B = BraidGroup(4)
sage: L = Link(B([-1, 3, 1, 3]))
sage: L.signature()
-1
sage: B = BraidGroup(8)
sage: L = Link(B([-2, 4, 1, 6, 1, 4]))
sage: L.signature()
-2
sage: L = Link(B([1, 2, 1, 2]))
sage: L.signature()
-2

writhe()¶

Return the writhe of self.

EXAMPLES:

sage: L = Link([[[1, -2, 3, -4, 2, -1, 4, -3]],[1, 1, -1, -1]])
sage: L.writhe()
0
sage: L = Link([[[-1, 2, -3, 4, 5, 1, -2, 6, 7, 3, -4, -7, -6,-5]],
....:            [-1, -1, -1, -1, 1, -1, 1]])
sage: L.writhe()
-3
sage: L = Link([[[-1, 2, 3, -4, 5, -6, 7, 8, -2, -5, 6, 1, -8, -3, 4, -7]],
....:            [-1, -1, -1, -1, 1, 1, -1, 1]])
sage: L.writhe()
-2

Links¶

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