#17030: Knot Theory as a part of GSoC 2014.
-------------------------------------+-------------------------------------
Reporter: amitjamadagni | Owner: amitjamadagni
Type: enhancement | Status: needs_work
Priority: major | Milestone: sage-6.9
Component: algebraic | Resolution:
topology | Merged in:
Keywords: | Reviewers: Miguel Marco, Karl-
Authors: Amit Jamadagni, | Dieter Crisman, Frédéric Chapoton
Miguel Marco | Work issues:
Report Upstream: N/A | Commit:
Branch: | cb84cec88f012844ac65d102296644a77f89818c
public/ticket/17030 | Stopgaps:
Dependencies: |
-------------------------------------+-------------------------------------
Comment (by jhpalmieri):
Hi Travis,
Please go ahead with a full review. I think a new `Knot` class would be
good. For future work, it would also be nice to have a catalog of links
and knots (e.g., `knots.Trefoil()`) and also ways of producing new links
from old. Could you add a `.. todo` list?
I agree that more documentation would be great, with definitions,
references, and (for `plot` at least) algorithms.
How is equality determined? How should it be? Should the oriented Gauss
code be good enough?
{{{
sage: trefoil = Link([[1, 5, 2, 4], [5, 3, 6, 2], [3, 1, 4, 6]])
sage: trefoil == Link([[1, 5, 2, 4], [5, 3, 6, 2], [3, 1, 4, 6]])
False
}}}
I haven't actually looked at the code, and I don't know if I ever will.
Some suggestions for miscellaneous minor cleanup:
{{{
#!diff
diff --git a/src/sage/knots/link.py b/src/sage/knots/link.py
index 04facf8..89f9624 100644
--- a/src/sage/knots/link.py
+++ b/src/sage/knots/link.py
@@ -10,7 +10,7 @@ A link is an embedding of one or more copies of
`\\mathbb{S}^1` in `\\mathbb{S}^
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 `\\mathbb{R}^2`
+Generically, the projection of a link to `\\Bold{R}^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
@@ -40,16 +40,12 @@ AUTHORS:
from sage.matrix.constructor import matrix
from sage.rings.integer_ring import ZZ
-from sage.rings.polynomial.polynomial_ring_constructor import
PolynomialRing
from sage.rings.finite_rings.integer_mod import Mod
from sage.graphs.digraph import DiGraph
from sage.graphs.graph import Graph
from copy import deepcopy, copy
from sage.rings.polynomial.laurent_polynomial_ring import
LaurentPolynomialRing
-from sage.rings.integer_ring import IntegerRing
-from sage.combinat.permutation import Permutations
-from sage.rings.finite_rings.integer_mod_ring import IntegerModRing
-from sage.symbolic.ring import SR, var
+from sage.symbolic.ring import SR
from sage.rings.integer import Integer
from sage.numerical.mip import MixedIntegerLinearProgram
from sage.functions.generalized import sign
@@ -74,7 +70,7 @@ class Link:
sage: L
Link with 2 components represented by 5 crossings
- Note that the strands of the braid that have no crossings at all are
ignored.
+ Note that the strands of the braid that have no crossings at all
are ignored.
- Oriented Gauss Code:
@@ -82,7 +78,7 @@ class Link:
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 wheter we
cross them
+ 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::
@@ -91,7 +87,7 @@ class Link:
sage: L
Knot represented by 8 crossings
- For links there is more than one component and the input is as
follows::
+ 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
@@ -177,15 +173,13 @@ class Link:
self._PD_code = None
else:
- raise Exception("Invalid Input: Data must be either a
list or a Braid")
+ raise ValueError("Invalid Input: Data must be either a
list or a Braid")
def __repr__(self):
"""
Return a string representation.
- OUTPUT:
-
- - A string.
+ OUTPUT: string representation
EXAMPLES::
@@ -211,9 +205,7 @@ class Link:
"""
Return the braid representation of the link.
- OUTPUT:
-
- - Braid representation of the link.
+ OUTPUT: braid representation of the link
EXAMPLES::
@@ -397,7 +389,7 @@ class Link:
b. The orientation of each crossing
- The following orientation was taken into consideration for
consturction
+ The following orientation was taken into consideration for
construction
of knots:
From the outgoing of the overcrossing if we move in the clockwise
direction
@@ -411,11 +403,9 @@ class Link:
The order of the orientation is same as the ordering of the
crossings in the
gauss code.
- Convention : under is denoted by -1, and over by +1 in the
crossing info.
+ Convention: under is denoted by -1, and over by +1 in the
crossing info.
- OUTPUT:
-
- - Oriented gauss code of the link
+ OUTPUT: Oriented gauss code of the link
EXAMPLES::
@@ -469,21 +459,19 @@ class Link:
def PD_code(self):
"""
- Return the Planar Diagram code of the link. The Planar Diagram is
returned
+ Return the planar diagram code of the link. 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.
- Suppose if the crossing is given by [a, b, c, d], then we
interpret this
- information as :
+ If the crossing is given by [a, b, c, d], then we interpret this
+ information as:
1. a is the entering component of the undercrossing
2. b, d are the components of the overcrossing
3. c is the leaving component of the undercrossing
- OUTPUT:
-
- - Planar Diagram representation of the link.
+ OUTPUT: planar diagram representation of the link.
EXAMPLES::
@@ -564,18 +552,16 @@ class Link:
def gauss_code(self):
"""
- Return the gauss_code of the link. Gauss code is generated by the
+ Return the Gauss code of the link. Gauss code is generated by the
following procedure:
a. Number the crossings from 1 to n.
b. Select a point on the knot and start moving along the
component.
c. At each crossing, take the number of the crossing, along with
- sign, which is '-' if it is a undercrossing and '+' if it is a
+ sign, which is '-' if it is an undercrossing and '+' if it is
an
overcrossing.
- OUTPUT:
-
- - Gauss code representation of the link.
+ OUTPUT: Gauss code representation of the link
EXAMPLES::
@@ -594,7 +580,7 @@ class Link:
def dt_code(self):
"""
- Return the dt_code of the knot. DT code is generated by the
following way.
+ Return the DT code of the knot. DT code is generated as follows.
Start moving along the knot, as we encounter the crossings we
start numbering them, so every crossing has two numbers assigned
to
@@ -605,10 +591,7 @@ class Link:
Take the even number with a negative sign if it is an
overcrossing
that we are encountering.
- OUTPUT:
-
- - DT Code representation of the knot. This is implemented only
- for knots.
+ OUTPUT: DT Code representation of the knot. This is implemented
only for knots.
EXAMPLES::
@@ -644,7 +627,7 @@ class Link:
crossing = i
break
if(label[2 * crossing + next_label % 2] == 1):
- raise Exception("Implemented only for knots")
+ raise ValueError("Implemented only for knots")
else:
label[2 * crossing + next_label % 2] = next_label
next_label = next_label + 1
@@ -674,15 +657,12 @@ class Link:
def _dowker_notation_(self):
"""
Return the dowker notation of the link. Similar to the PD Code we
number the
- components. So every crossing is represented by four numbers. We
focus on
- the incoming entites of the under and the over crossing. It is
the pair of incoming
- under cross and the incoming over cross. This information at
every cross
+ 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
+ undercross and the incoming overcross. This information at every
cross
gives the dowker notation.
- OUTPUT:
-
- - List containing the pair of incoming under cross and the
incoming
- over cross.
+ OUTPUT: List containing the pair of incoming undercross and the
incoming overcross.
EXAMPLES::
@@ -705,16 +685,14 @@ class Link:
def _braidwordcomponents_(self):
"""
- Return the disjoint braid components, if any, else returns the
braid itself.
- For example consider the braid [-1, 3, 1, 3] this can be viewed
as a braid
+ Return the disjoint braid components, if any, else return the
braid itself.
+ For example consider the braid [-1, 3, 1, 3]. This can be viewed
as a braid
with components as [-1, 1] and [3, 3]. There is no common
crossing to these
two (in sense there is a crossing between strand 1 and 2,
crossing between
3 and 4 but no crossing between strand 2 and 3,so these can be
viewed as
independent components in the braid).
- OUTPUT:
-
- - List containing the components is returned
+ OUTPUT: List containing the components is returned
EXAMPLES::
@@ -733,7 +711,7 @@ class Link:
b = self.braid()
ml = list(b.Tietze())
if ml == []:
- raise Exception("The braid remains the same with no
components")
+ raise ValueError("The braid remains the same with no
components")
else:
l = list(set([abs(k) for k in ml]))
missing1 = list(set(range(min(l), max(l) + 1)) - set(l))
@@ -755,11 +733,9 @@ class Link:
def _braidwordcomponentsvector_(self):
"""
- The list from the braidwordcomponents is flattened to give out
the vector form.
+ The list from :meth:`_braidwordcomponents_` is flattened to give
the vector form.
- OUTPUT:
-
- - Vector containing braidwordcomponents
+ OUTPUT: Vector containing :meth:`_braidwordcomponents_`
EXAMPLES::
@@ -780,14 +756,13 @@ class Link:
def _homology_generators_(self):
"""
- The set of generators for the first homology group of the
connected Seifert surface of the given link.
- This method uses the braidwordcomponentsvector to generate the
homology generators.
- The position of the repeated element w.r.t the
braidwordcomponentvector list is
- compiled into a list.
-
- OUTPUT:
+ The set of generators for the first homology group of the
+ connected Seifert surface of the given link. This method uses
+ :meth:`_braidwordcomponentsvector_` to generate the homology
+ generators. The position of the repeated element w.r.t.
+ :meth:`_braidwordcomponentsvector_` is compiled into a list.
- - The homology generators relating to the braid word
representation
+ OUTPUT: The homology generators relating to the braid word
representation
EXAMPLES::
@@ -802,6 +777,7 @@ class Link:
sage: L = Link(B([-2, 4, 1, 6, 1, 4]))
sage: L._homology_generators_()
[0, 2, 0, 4, 0]
+
"""
x4 = self._braidwordcomponentsvector_()
hom_gen = []
@@ -819,10 +795,8 @@ class Link:
"""
Return the Seifert Matrix associated with the link.
- OUTPUT:
-
- - The intersection matrix of a (not necessarily minimal) Seifert
surface of the
- link.
+ OUTPUT: The intersection matrix of a (not necessarily minimal)
Seifert
+ surface of the link.
EXAMPLES::
@@ -892,9 +866,7 @@ class Link:
"""
Return the number of connected components of the link.
- OUTPUT:
-
- - Connected components of the link
+ OUTPUT: Connected components of the link
EXAMPLES::
@@ -919,12 +891,10 @@ class Link:
def is_knot(self):
"""
- Return True if the link is knot.
+ Return True if the link is a knot.
Every knot is a link but the converse is not true.
- OUTPUT:
-
- - True if knot else False
+ OUTPUT: True if knot else False
EXAMPLES::
@@ -946,9 +916,7 @@ class Link:
"""
Return the genus of the link
- OUTPUT:
-
- - Genus of the Link
+ OUTPUT: genus of the link
EXAMPLES::
@@ -1004,9 +972,7 @@ class Link:
"""
Return the signature of the link
- OUTPUT:
-
- - Signature of the Link
+ OUTPUT: signature of the link
EXAMPLES::
@@ -1039,9 +1005,7 @@ class Link:
- ``var`` -- (default: ``'t'``); the variable in the polynomial.
- OUTPUT:
-
- - Alexander Polynomial of the Link
+ OUTPUT: Alexander Polynomial of the Link
EXAMPLES::
@@ -1067,9 +1031,7 @@ class Link:
"""
Return the determinant of the knot
- OUTPUT:
-
- - Determinant of the Knot
+ OUTPUT: determinant of the knot
EXAMPLES::
@@ -1089,15 +1051,13 @@ class Link:
a = self.alexander_polynomial()
return Integer(abs(a(-1)))
else:
- raise Exception("Determinant implmented only for knots")
+ raise ValueError("Determinant implemented only for knots")
def arf_invariant(self):
"""
Return the arf invariant. Arf invariant is defined only for
knots.
- OUTPUT:
-
- - Arf invariant of knot
+ OUTPUT: Arf invariant of the knot
EXAMPLES::
@@ -1120,20 +1080,16 @@ class Link:
else:
return 1
else:
- raise Exception("Arf invariant is defined only for knots")
+ raise ValueError("Arf invariant is defined only for knots")
def is_alternating(self):
"""
- Return True if the given knot diagram is alternating else returns
False.
- Alternating diagram implies every over cross is followed by an
under cross
+ Return True if the given knot diagram is alternating else return
False.
+ Alternating diagram means that every over cross is followed by an
under cross
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.
-
- OUTPUT:
-
- - True if the knot diagram is alternating else False
+ In the gauss code, if the sign is alternating, True is returned,
else
+ the knot is not alternating so False is returned.
EXAMPLES::
@@ -1169,10 +1125,6 @@ class Link:
"""
Return the orientation of the crossings of the link diagram.
- OUTPUT:
-
- - Orientation of the crossings.
-
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]])
@@ -1201,15 +1153,11 @@ class Link:
def seifert_circles(self):
"""
- Return the seifert circles from the link diagram. Seifert circles
are the circles
+ Return the Seifert circles from the link diagram. 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.
- OUTPUT:
-
- - Seifert circles of the given link diagram.
-
EXAMPLES::
sage: L = Link([[[1, -2, 3, -4, 2, -1, 4, -3]],[1, 1, -1,
-1]])
@@ -1263,9 +1211,7 @@ class Link:
its boundary, with a sign deppending on the orientation of the
segment
as part of the boundary.
- OUTPUT:
-
- - Regions of the knot.
+ OUTPUT: regions of the knot
EXAMPLES::
@@ -1344,9 +1290,7 @@ class Link:
"""
Return the writhe of the knot.
- OUTPUT:
-
- - Writhe of the knot.
+ OUTPUT: writhe of the knot
EXAMPLES::
@@ -1367,16 +1311,14 @@ class Link:
def jones_polynomial(self, var='q'):
"""
- Return the jones polynomial of the link.
+ Return the Jones polynomial of the link.
INPUT:
- ``var`` -- (default: ``'q'``); the variable in the polynomial.
- OUTPUT:
-
- - Jones Polynomial of the link. It is a polynomial in var, as an
element
- of the symbolic ring.
+ OUTPUT: Jones polynomial of the link. It is a polynomial in
+ ``var``, as an element of the symbolic ring.
EXAMPLES::
@@ -1394,6 +1336,7 @@ class Link:
sage: L = Link(l5)
sage: L.jones_polynomial()
-q^(3/2) + sqrt(q) - 2/sqrt(q) + 1/q^(3/2) - 2/q^(5/2) +
1/q^(7/2)
+
"""
t = SR(var)
poly = self._bracket_(t)
@@ -1420,8 +1363,6 @@ class Link:
sage: L = Link([[2, 1, 3, 4], [4, 3, 1, 2]])
sage: L._bracket_()
-q^4 - 1/q^4
-
-
"""
t = SR(variable)
if len(self.PD_code()) == 1:
@@ -1482,7 +1423,6 @@ class Link:
sage: L = Link([[1, 1, 2, 2], [3, 3, 4, 4]])
sage: L._isolated_components_()
[[[1, 1, 2, 2]], [[3, 3, 4, 4]]]
-
"""
G = Graph()
for c in self.PD_code():
}}}
--
Ticket URL: <http://trac.sagemath.org/ticket/17030#comment:117>
Sage <http://www.sagemath.org>
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