Trac #33969: Implement missing KnotInfo wrappers for polynomial invar…

…iants

A couple of wrappers for link properties listed in the
[https://knotinfo.math.indiana.edu/ KnotInfo] and
[https://linkinfo.sitehost.iu.edu/ LinkInfo] databases have already been
implemented in #30352. But since there are more than 120 of them there
are still a lot missing. Here we add missing polynomial link invariants,
explicitly the Conway and Khovanov polynomials.

Furthermore we let Sage point to the current version of
[https://pypi.org/project/database-knotinfo/ database_knotinfo].

URL: https://trac.sagemath.org/33969
Reported by: soehms
Ticket author(s): Sebastian Oehms
Reviewer(s): Matthias Koeppe

build/pkgs/database_knotinfo/checksums.ini

@@ -1,5 +1,5 @@
 tarball=database_knotinfo-VERSION.tar.gz
-sha1=187e6b5ee2a935e3a50bc7648b181dfc7cb7bfa2
-md5=90822e09a1a84c8dbb84e20773c367f1
-cksum=1855405219
+sha1=16039d4e399efc78e4b1278527019f4bcdfdde13
+md5=3095993756f6b51d14c35adae5a75930
+cksum=2884062991
 upstream_url=https://pypi.io/packages/source/d/database_knotinfo/database_knotinfo-VERSION.tar.gz

build/pkgs/database_knotinfo/package-version.txt

@@ -1 +1 @@
-2021.10.1
+2022.7.1

src/sage/databases/knotinfo_db.py

@@ -75,6 +75,7 @@ class KnotInfoColumns(Enum):
          'PD Notation (vector)',
          'PD Notation (KnotTheory)',
          'Braid Notation',
+         'Quasipositive Braid',
          'Multivariable Alexander Polynomial',
          'HOMFLYPT Polynomial',
          'Unoriented',
@@ -822,6 +823,8 @@ def _test_database(self, **options):
     'homfly_polynomial':    ['HOMFLY',               KnotInfoColumnTypes.OnlyKnots],
     'homflypt_polynomial':  ['HOMFLYPT Polynomial',  KnotInfoColumnTypes.OnlyLinks],
     'kauffman_polynomial':  ['Kauffman',             KnotInfoColumnTypes.KnotsAndLinks],
+    'khovanov_polynomial':  ['Khovanov',             KnotInfoColumnTypes.KnotsAndLinks],
+    'khovanov_torsion_polynomial': ['Khovanov Torsion', KnotInfoColumnTypes.OnlyKnots],
     'determinant':          ['Determinant',          KnotInfoColumnTypes.KnotsAndLinks],
     'positive':             ['Positive',             KnotInfoColumnTypes.OnlyKnots],
     'fibered':              ['Fibered',              KnotInfoColumnTypes.OnlyKnots],
@@ -1089,5 +1092,60 @@ def _test_database(self, **options):
         '1-3*t+ 3*t^2-3*t^3+ t^4',
         '1-3*t+ 5*t^2-3*t^3+ t^4',
         '1-t+ t^2-t^3+ t^4-t^5+ t^6',
-        '3-5*t+ 3*t^2']
+        '3-5*t+ 3*t^2'],
+    dc.conway_polynomial: [
+        '1',
+        '1+z^2',
+        '1-z^2',
+        '1+3*z^2+z^4',
+        '1+2*z^2',
+        '1-2*z^2',
+        '1-z^2-z^4',
+        '1+z^2+z^4',
+        '1+6*z^2+5*z^4+z^6',
+        '1+3*z^2',
+        '-z',
+        'z',
+        '-2*z',
+        '2*z + z^3',
+        'z^3',
+        'z^3',
+        '-2*z + z^3',
+        '2*z + 2*z^3',
+        '-3*z-2*z^3',
+        '3*z + 2*z^3',
+        '-3*z-4*z^3-z^5'],
+    dc.khovanov_polynomial: [
+        '',
+        'q^(-9)t^(-3)+q^(-5)t^(-2)+q^(-3)+q^(-1)',
+        'q^(-5)t^(-2)+q^(-1)t^(-1)+q+q^(-1)+qt+q^5t^2',
+        'q^(-15)t^(-5)+q^(-11)t^(-4)+q^(-11)t^(-3)+q^(-7)t^(-2)+q^(-5)+q^(-3)',
+        'q^(-13)t^(-5)+q^(-9)t^(-4)+q^(-9)t^(-3)+(q^(-7)+q^(-5))t^(-2)+q^(-3)t^(-1)+q^(-3)+q^(-1)',
+        'q^(-9)t^(-4)+q^(-5)t^(-3)+q^(-5)t^(-2)+(q^(-3)+q^(-1))t^(-1)+2q+q^(-1)+qt+q^5t^2',
+        'q^(-11)t^(-4)+(q^(-9)+q^(-7))t^(-3)+(q^(-7)+q^(-5))t^(-2)+(q^(-5)+q^(-3))t^(-1)+q^(-3)+2q^(-1)+tq^(-1)+q^3t^2',
+        'q^(-7)t^(-3)+(q^(-5)+q^(-3))t^(-2)+(q^(-3)+q^(-1))t^(-1)+2q+2q^(-1)+t(q+q^3)+(q^3+q^5)t^2+q^7t^3',
+        'q^(-21)t^(-7)+q^(-17)t^(-6)+q^(-17)t^(-5)+q^(-13)t^(-4)+q^(-13)t^(-3)+q^(-9)t^(-2)+q^(-7)+q^(-5)',
+        'q^(-17)t^(-7)+q^(-13)t^(-6)+q^(-13)t^(-5)+(q^(-11)+q^(-9))t^(-4)+(q^(-9)+q^(-7))t^(-3)+(q^(-7)+q^(-5))t^(-2)+q^(-3)t^(-1)+q^(-3)+q^(-1)',
+        '1 + q^(-2) + 1/(q^6*t^2) + 1/(q^4*t^2)',
+        '1 + q^2 + q^4*t^2 + q^6*t^2',
+        '1 + q^(-2) + 1/(q^10*t^4) + 1/(q^8*t^4) + 1/(q^6*t^2) + 1/(q^2*t)',
+        'q^2 + q^4 + q^6*t^2 + q^10*t^3 + q^10*t^4 + q^12*t^4',
+        '2 + 2/q^2 + 1/(q^8*t^3) + 1/(q^6*t^2) + 1/(q^4*t^2) + 1/(q^2*t) + t + q^4*t^2',
+        '2 + 2/q^2 + 1/(q^8*t^3) + 1/(q^6*t^2) + 1/(q^4*t^2) + 1/(q^2*t) + t + q^4*t^2',
+        '1 + 2/q^2 + 1/(q^10*t^4) + 1/(q^8*t^4) + 1/(q^8*t^3) + 2/(q^6*t^2) + 1/(q^4*t^2) + 2/(q^2*t) + t + q^2*t + q^4*t^2',
+        'q^2 + q^4 + q^4*t + 2*q^6*t^2 + q^8*t^2 + 2*q^10*t^3 + 2*q^10*t^4 + q^12*t^4 + q^12*t^5 + q^14*t^5 + q^16*t^6',
+        'q^(-4) + q^(-2) + 1/(q^16*t^6) + 1/(q^14*t^6) + 1/(q^14*t^5) + 1/(q^12*t^4) + 1/(q^10*t^4) + 1/(q^10*t^3) + 1/(q^8*t^3) + 1/(q^8*t^2) + 1/(q^6*t^2) + 1/(q^4*t)',
+        'q^2 + q^4 + q^4*t + q^6*t^2 + q^8*t^2 + q^8*t^3 + q^10*t^3 + q^10*t^4 + q^12*t^4 + q^14*t^5 + q^14*t^6 + q^16*t^6',
+        'q^(-6) + q^(-4) + 1/(q^18*t^6) + 1/(q^16*t^6) + 1/(q^16*t^5) + 1/(q^12*t^4) + 1/(q^12*t^3) + 1/(q^8*t^2)'],
+    dc.khovanov_torsion_polynomial: [
+        '',
+        'Q^(-7)t^(-2)',
+        'Q^(-3)t^(-1)+Q^3t^2',
+        'Q^(-13)t^(-4)+Q^(-9)t^(-2)',
+        'Q^(-11)t^(-4)+Q^(-7)t^(-2)+Q^(-5)t^(-1)',
+        'Q^(-7)t^(-3)+Q^(-3)t^(-1)+Q^(-1)+Q^3t^2',
+        'Q^(-9)t^(-3)+Q^(-7)t^(-2)+Q^(-5)t^(-1)+Q^(-3)+Qt^2',
+        'Q^(-5)t^(-2)+Q^(-3)t^(-1)+Q^(-1)+Qt+Q^3t^2+Q^5t^3',
+        'Q^(-19)t^(-6)+Q^(-15)t^(-4)+Q^(-11)t^(-2)',
+        'Q^(-15)t^(-6)+Q^(-11)t^(-4)+Q^(-9)t^(-3)+Q^(-7)t^(-2)+Q^(-5)t^(-1)']
 }

src/sage/knots/knotinfo.py

@@ -216,6 +216,7 @@
 AUTHORS:
 
 - Sebastian Oehms August 2020: initial version
+- Sebastian Oehms June   2022: add :meth:`conway_polynomial` and :meth:`khovanov_polynomial` (:trac:`33969`)
 
 Thanks to Chuck Livingston and Allison Moore for their support. For further acknowledgments see the correspondig hompages.
 """
@@ -1620,6 +1621,161 @@ def alexander_polynomial(self, var='t', original=False, laurent_poly=False):
         exp = ap.exponents()
         return t ** ((-max(exp) - min(exp)) // 2) * ap
 
+    @cached_method
+    def conway_polynomial(self, var='t', original=False):
+        r"""
+        Return the Conway polynomial according to the value of column
+        ``conway_polynomial`` for this knot or link as an instance of
+        :class:`~sage.rings.polynomial.polynomial_element.Polynomial`.
+
+        It is obtained from the Seifert matrix `V` of ``self`` by the following
+        formula (see the KnotInfo description web-page; to launch it see the
+        example below):
+
+        .. MATH::
+
+            \nabla(L) = \det(t^{\frac{1}{2}} V -t^{\frac{-1}{2}} V^t)
+
+        Here `V^t` stands for the transpose of `V`.
+
+
+        INPUT:
+
+        - ``var`` -- (default: ``'t'``) the variable
+        - ``original`` -- boolean (optional, default ``False``) if set to
+          ``True`` the original table entry is returned as a string
+
+        OUTPUT:
+
+        A polynomial over the integers, more precisely an instance of
+        :class:`~sage.rings.polynomial.polynomial_element.Polynomial`.
+        If ``original`` is set to ``True`` then a string is returned.
+
+        EXAMPLES::
+
+            sage: from sage.knots.knotinfo import KnotInfo
+            sage: K = KnotInfo.K4_1
+            sage: Kc = K.conway_polynomial(); Kc
+            -t^2 + 1
+            sage: L = KnotInfo.L5a1_0
+            sage: Lc = L.conway_polynomial(); Lc
+            t^3
+
+        Comparision to Sage's results::
+
+            sage: Kc == K.link().conway_polynomial()
+            True
+            sage: Lc == L.link().conway_polynomial()
+            True
+
+        Launch the KnotInfo description web-page::
+
+            sage: K.items.conway_polynomial.description_webpage() # not tested
+            True
+        """
+        conway_polynomial = self[self.items.conway_polynomial]
+
+        if original:
+            return conway_polynomial
+
+        from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
+        R = PolynomialRing(ZZ, var)
+
+        if not conway_polynomial and self.crossing_number() == 0:
+            return R.one()
+
+        t, = R.gens()
+        lc = {'z':  t}
+        return R(eval_knotinfo(conway_polynomial, locals=lc))
+
+    @cached_method
+    def khovanov_polynomial(self, var1='q', var2='t', base_ring=ZZ, original=False):
+        r"""
+        Return the Khovanov polynomial according to the value of column
+        ``khovanov_polynomial`` for this knot or link as an instance of
+        :class:`~sage.rings.polynomial.laurent_polynomial.LaurentPolynomial_mpair`.
+
+        INPUT:
+
+        - ``var1`` -- (default: ``'q'``) the first variable
+        - ``var2`` -- (default: ``'t'``) the second variable
+        - ``base_ring`` -- (default: ``ZZ``) the ring of the polynomial's
+          coefficients
+        - ``original`` -- boolean (optional, default ``False``) if set to
+          ``True`` the original table entry is returned as a string
+
+        OUTPUT:
+
+        A Laurent polynomial over the integers, more precisely an instance of
+        :class:`~sage.rings.polynomial.laurent_polynomial.LaurentPolynomial_mpair`.
+        If ``original`` is set to ``True`` then a string is returned.
+
+        .. NOTE ::
+
+            The Khovanov polynomial given in KnotInfo corresponds to the mirror
+            image of the given knot for a `list of 140 exceptions
+            <https://raw.githubusercontent.com/soehms/database_knotinfo/main/hints/list_of_mirrored_khovanov_polynonmial.txt>`__.
+
+        EXAMPLES::
+
+            sage: from sage.knots.knotinfo import KnotInfo
+            sage: K = KnotInfo.K6_3
+            sage: Kk = K.khovanov_polynomial(); Kk
+            q^7*t^3 + q^5*t^2 + q^3*t^2 + q^3*t + q*t + 2*q + 2*q^-1 + q^-1*t^-1
+            + q^-3*t^-1 + q^-3*t^-2 + q^-5*t^-2 + q^-7*t^-3
+            sage: Kk2 = K.khovanov_polynomial(var1='p', base_ring=GF(2)); Kk2
+            p^7*t^3 + p^5*t^3 + p^5*t^2 + p^3*t + p^-1 + p^-1*t^-1 + p^-3*t^-2 + p^-7*t^-3
+
+            sage: L = KnotInfo.L5a1_0
+            sage: Lk = L.khovanov_polynomial(); Lk
+            q^4*t^2 + t + 2 + 2*q^-2 + q^-2*t^-1 + q^-4*t^-2 + q^-6*t^-2 + q^-8*t^-3
+
+        Comparision to Sage's results::
+
+            sage: Kk == K.link().khovanov_polynomial()
+            True
+            sage: Kk2 == K.link().khovanov_polynomial(var1='p', base_ring=GF(2))
+            True
+            sage: Lk == L.link().khovanov_polynomial()
+            True
+        """
+        khovanov_polynomial = self[self.items.khovanov_polynomial]
+
+        if original:
+            return khovanov_polynomial
+
+        from sage.rings.polynomial.laurent_polynomial_ring import LaurentPolynomialRing
+        var_names = [var1, var2]
+        R = LaurentPolynomialRing(base_ring, var_names)
+
+        if not khovanov_polynomial and self.crossing_number() == 0:
+            return R({(1, 0): 1, (-1, 0): 1})
+
+        ch = base_ring.characteristic()
+        if ch == 2:
+            if not self.is_knot():
+                raise NotImplementedError('Khovanov polynomial available only for knots in characteristic 2')
+            khovanov_torsion_polynomial = self[self.items.khovanov_torsion_polynomial]
+            khovanov_torsion_polynomial = khovanov_torsion_polynomial.replace('Q', 'q')
+            khovanov_polynomial = '%s + %s' % (khovanov_polynomial, khovanov_torsion_polynomial)
+
+        if not khovanov_polynomial:
+            # given just for links with less than 12 crossings
+            raise NotImplementedError('Khovanov polynomial not available for this link')
+
+        from sage.repl.preparse import implicit_mul
+        # since implicit_mul does not know about the choice of variable names
+        # we have to insert * between them separately
+        for i in ['q', 't',')']:
+            for j in ['q', 't', '(']:
+                khovanov_polynomial = khovanov_polynomial.replace('%s%s' % (i, j), '%s*%s' % (i, j))
+        khovanov_polynomial = implicit_mul(khovanov_polynomial)
+        gens = R.gens_dict()
+        lc = {}
+        lc['q'] = gens[var1]
+        lc['t'] = gens[var2]
+
+        return R(eval_knotinfo(khovanov_polynomial, locals=lc))
 
     @cached_method
     def link(self, use_item=db.columns().pd_notation, snappy=False):