Free monoid generated by prime knots available via the¶

KnotInfoBase class.

A generator of this free abelian monoid is a prime knot according to the list at KnotInfo. A fully amphicheiral prime knot is represented by exactly one generator with the corresponding name. For non-chiral prime knots, there are additionally one or three generators with the suffixes m, r and c which specify the mirror and reverse images according to their symmetry type.

AUTHORS:

Sebastian Oehms June 2024: initial version

class sage.knots.free_knotinfo_monoid.FreeKnotInfoMonoid(max_crossing_number, category=None, prefix=None, **kwds)[source]¶

Bases: IndexedFreeAbelianMonoid

Initialize self with generators belonging to prime knots with at most max_crossing_number crossings.

Element[source]¶: alias of FreeKnotInfoMonoidElement

from_knot(knot, unique=True)[source]¶

Create an element of this abelian monoid from knot.

INPUT:

knot – an instance of Knot
unique – boolean (default is True). This only affects the case where a unique identification is not possible. If set to False you can obtain a matching list (see explanation of the output below)

OUTPUT:

An instance of the element class of self per default. If the keyword argument unique then a list of such instances is returned.

EXAMPLES:

Sage

sage: from sage.knots.free_knotinfo_monoid import FreeKnotInfoMonoid
sage: FKIM =  FreeKnotInfoMonoid()
sage: K = KnotInfo.K5_1.link().mirror_image()
sage: FKIM.from_knot(K)
KnotInfo['K5_1m']

sage: # optional - database_knotinfo
sage: K = Knot(KnotInfo.K9_12.braid())
sage: FKIM.from_knot(K)                   # long time
Traceback (most recent call last):
...
NotImplementedError: this (possibly non prime) knot cannot be
identified uniquely by KnotInfo
use keyword argument `unique` to obtain more details
sage: FKIM.from_knot(K, unique=False)     # long time
[KnotInfo['K4_1']*KnotInfo['K5_2'], KnotInfo['K9_12']]

Python

>>> from sage.all import *
>>> from sage.knots.free_knotinfo_monoid import FreeKnotInfoMonoid
>>> FKIM =  FreeKnotInfoMonoid()
>>> K = KnotInfo.K5_1.link().mirror_image()
>>> FKIM.from_knot(K)
KnotInfo['K5_1m']

>>> # optional - database_knotinfo
>>> K = Knot(KnotInfo.K9_12.braid())
>>> FKIM.from_knot(K)                   # long time
Traceback (most recent call last):
...
NotImplementedError: this (possibly non prime) knot cannot be
identified uniquely by KnotInfo
use keyword argument `unique` to obtain more details
>>> FKIM.from_knot(K, unique=False)     # long time
[KnotInfo['K4_1']*KnotInfo['K5_2'], KnotInfo['K9_12']]

Sage Live

from sage.knots.free_knotinfo_monoid import FreeKnotInfoMonoid
FKIM =  FreeKnotInfoMonoid()
K = KnotInfo.K5_1.link().mirror_image()
FKIM.from_knot(K)
# optional - database_knotinfo
K = Knot(KnotInfo.K9_12.braid())
FKIM.from_knot(K)                   # long time
FKIM.from_knot(K, unique=False)     # long time

inject_variables(select=None, verbose=True)[source]¶

Inject self with its name into the namespace of the Python code from which this function is called.

INPUT:

select – instance of KnotInfoBase, KnotInfoSeries or an integer. In all cases the input is used to restrict the injected generators to the according subset (number of crossings in the case of integer)
verbose – boolean (optional, default True) to suppress the message printed on the invocation

EXAMPLES:

Sage

sage: from sage.knots.free_knotinfo_monoid import FreeKnotInfoMonoid
sage: FKIM = FreeKnotInfoMonoid(5)
sage: FKIM.inject_variables(select=3)
Defining K3_1
Defining K3_1m
sage: FKIM.inject_variables(select=KnotInfo.K5_2)
Defining K5_2
Defining K5_2m
sage: FKIM.inject_variables(select=KnotInfo.K5_2.series())
Defining K5_1
Defining K5_1m
sage: FKIM.inject_variables()
Defining K0_1
Defining K4_1

Python

>>> from sage.all import *
>>> from sage.knots.free_knotinfo_monoid import FreeKnotInfoMonoid
>>> FKIM = FreeKnotInfoMonoid(Integer(5))
>>> FKIM.inject_variables(select=Integer(3))
Defining K3_1
Defining K3_1m
>>> FKIM.inject_variables(select=KnotInfo.K5_2)
Defining K5_2
Defining K5_2m
>>> FKIM.inject_variables(select=KnotInfo.K5_2.series())
Defining K5_1
Defining K5_1m
>>> FKIM.inject_variables()
Defining K0_1
Defining K4_1

Sage Live

from sage.knots.free_knotinfo_monoid import FreeKnotInfoMonoid
FKIM = FreeKnotInfoMonoid(5)
FKIM.inject_variables(select=3)
FKIM.inject_variables(select=KnotInfo.K5_2)
FKIM.inject_variables(select=KnotInfo.K5_2.series())
FKIM.inject_variables()

class sage.knots.free_knotinfo_monoid.FreeKnotInfoMonoidElement(F, x)[source]¶

Bases: IndexedFreeAbelianMonoidElement

An element of an indexed free abelian monoid.

as_knot()[source]¶

Return the knot represented by self.

EXAMPLES:

Sage

sage: from sage.knots.free_knotinfo_monoid import FreeKnotInfoMonoid
sage: FKIM =  FreeKnotInfoMonoid()
sage: FKIM.inject_variables(select=3)
Defining K3_1
Defining K3_1m
sage: K = K3_1^2 * K3_1m
sage: K.as_knot()
Knot represented by 9 crossings

Python

>>> from sage.all import *
>>> from sage.knots.free_knotinfo_monoid import FreeKnotInfoMonoid
>>> FKIM =  FreeKnotInfoMonoid()
>>> FKIM.inject_variables(select=Integer(3))
Defining K3_1
Defining K3_1m
>>> K = K3_1**Integer(2) * K3_1m
>>> K.as_knot()
Knot represented by 9 crossings

Sage Live

from sage.knots.free_knotinfo_monoid import FreeKnotInfoMonoid
FKIM =  FreeKnotInfoMonoid()
FKIM.inject_variables(select=3)
K = K3_1^2 * K3_1m
K.as_knot()

to_knotinfo()[source]¶

Return a word representing self as a list of pairs.

Each pair (ki, sym) consists of a KnotInfoBase instance ki and SymmetryMutant instance sym.

EXAMPLES:

Sage

sage: from sage.knots.free_knotinfo_monoid import FreeKnotInfoMonoid
sage: FKIM =  FreeKnotInfoMonoid()
sage: FKIM.inject_variables(select=3)
Defining K3_1
Defining K3_1m
sage: K = K3_1^2 * K3_1m
sage: K.to_knotinfo()
[(<KnotInfo.K3_1: '3_1'>, <SymmetryMutant.itself: 's'>),
(<KnotInfo.K3_1: '3_1'>, <SymmetryMutant.itself: 's'>),
(<KnotInfo.K3_1: '3_1'>, <SymmetryMutant.mirror_image: 'm'>)]

Python

>>> from sage.all import *
>>> from sage.knots.free_knotinfo_monoid import FreeKnotInfoMonoid
>>> FKIM =  FreeKnotInfoMonoid()
>>> FKIM.inject_variables(select=Integer(3))
Defining K3_1
Defining K3_1m
>>> K = K3_1**Integer(2) * K3_1m
>>> K.to_knotinfo()
[(<KnotInfo.K3_1: '3_1'>, <SymmetryMutant.itself: 's'>),
(<KnotInfo.K3_1: '3_1'>, <SymmetryMutant.itself: 's'>),
(<KnotInfo.K3_1: '3_1'>, <SymmetryMutant.mirror_image: 'm'>)]

Sage Live

from sage.knots.free_knotinfo_monoid import FreeKnotInfoMonoid
FKIM =  FreeKnotInfoMonoid()
FKIM.inject_variables(select=3)
K = K3_1^2 * K3_1m
K.to_knotinfo()