Links¶
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.
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 figureeight 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
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 ofself
.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 thann
.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_ncoloring
 Chapter 3 of [Liv1993]
See also

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 variablevar2
– (default:'M'
) the second variablenormalization
– (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 figureeight 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 returnsFalse
.Alternating diagram implies every overcross is followed by an undercross or the viceversa.
We look at the Gauss code if the sign is alternating,
True
is returned else the knot is not alternatingFalse
is returned.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

is_colorable
(n)¶ Return whether the link is
n
colorable.A link is
n
colorable if its arcs can be painted withn
colours, labeled from0
ton  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 thann
.EXAMPLES:
We show that the trefoil knot is 3colorable:
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_ncoloring
 Chapter 3 of [Liv1993]
See also

is_knot
()¶ Return
True
ifself
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
isTrue
, 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
isNone
return a polynomial in the variable \(A\) or \(t\), depending on the valueskein_normalization
. In particular, ifskein_normalization
isFalse
, 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 ofself
to compute the polynomial of the trace closure of the braid, andstatesum
which recursively computes the Kauffman bracket ofself
. 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 isself
, 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 ringskein_normalization
– boolean (default:False
); determines the variable of the resulting polynomialalgorithm
– 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
ifFalse
, 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 invariab
.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 “KinoshitaTerasaka” 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(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 ringheight
– the height of the homology to compute, if not specified, all the heights are computeddegree
– 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 LevineTristram signature, the equivariant signature or the TristramLevine signature.
See also
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
 \(\omega\) – a complex number of modulus 1. This is assumed to be
coercible to

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 crossingscomponent_gap
– (default: 0.5) the gap between isolated componentssolver
– the linear solver to use, seeMixedIntegerLinearProgram
.color
– (default: ‘blue’) a color or a coloring (as returned bycolorings()
.
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 figureeight 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
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