Trac #27522: implement Conway polynomial

as deduced from Alexander polynomial

URL: https://trac.sagemath.org/27522
Reported by: chapoton
Ticket author(s): Frédéric Chapoton
Reviewer(s): Sebastian Oehms, Travis Scrimshaw

src/sage/knots/link.py

@@ -37,7 +37,7 @@
 
 - Miguel Angel Marco Buzunariz
 - Amit Jamadagni
-- Sebastian Oehms (October 2020, add :meth:`get_knotinfo` and meth:`is_isotopic`)
+- Sebastian Oehms (October 2020, add :meth:`get_knotinfo` and :meth:`is_isotopic`)
 """
 
 # ****************************************************************************
@@ -1895,6 +1895,8 @@ def alexander_polynomial(self, var='t'):
             sage: L = Link(B([-2, 4, 1, 6, 1, 4]))
             sage: L.alexander_polynomial()
             0
+
+        .. SEEALSO:: :meth:`conway_polynomial`
         """
         R = LaurentPolynomialRing(ZZ, var)
         # The Alexander polynomial of disjoint links are defined to be 0
@@ -1909,6 +1911,53 @@ def alexander_polynomial(self, var='t'):
             return t ** ((-max(exp) - min(exp)) // 2) * f
         return f
 
+    def conway_polynomial(self):
+        """
+        Return the Conway polynomial of ``self``.
+
+        This is closely related to the Alexander polynomial.
+
+        See :wikipedia:`Alexander_polynomial` for the definition.
+
+        EXAMPLES::
+
+            sage: B = BraidGroup(3)
+            sage: L = Link(B([1, -2, 1, -2]))
+            sage: L.conway_polynomial()
+            -t^2 + 1
+            sage: Link([[1, 5, 2, 4], [3, 9, 4, 8], [5, 1, 6, 10],
+            ....:       [7, 3, 8, 2], [9, 7, 10, 6]])
+            Link with 1 component represented by 5 crossings
+            sage: _.conway_polynomial()
+            2*t^2 + 1
+            sage: B = BraidGroup(4)
+            sage: L = Link(B([1,3]))
+            sage: L.conway_polynomial()
+            0
+
+        .. SEEALSO:: :meth:`alexander_polynomial`
+        """
+        alex = self.alexander_polynomial()
+        L = alex.parent()
+        R = L.polynomial_ring()
+        if alex == 0:
+            return R.zero()
+
+        t = L.gen()
+        alex = alex(t**2)
+        exp = alex.exponents()
+        alex = t**((-max(exp) - min(exp)) // 2) * alex
+
+        conway = R.zero()
+        t_poly = R.gen()
+        binom = t - ~t
+        while alex:
+            M = max(alex.exponents())
+            coeff = alex[M]
+            alex -= coeff * binom**M
+            conway += coeff * t_poly**M 
+        return conway
+
     def determinant(self):
         r"""
         Return the determinant of ``self``.