Link.alexander now handles inputs with R1 and R2 simplifications

Author: NathanDunfield

spherogram_src/links/alexander.py

@@ -249,9 +249,28 @@ def good_exhaustion(link, max_failed_tries=20):
     return E_best
 
 
+
 def alexander(K):
+    """
+    This function assumes no Type I moves is available.
+
+    Make sure the method that calls this gracefully handle inputs
+    where such moves are present.
+
+    >>> from .invariants import Link
+    >>> Link([(1, 4, 2, 1), (2, 4, 3, 3)]).alexander_polynomial()
+    1
+    >>> L = Link('K3a1')
+    >>> L.unlinked_unknot_components = 1
+    >>> L.alexander_polynomial()
+    0
+    """
     c = len(K.crossings)
-    if c < 100:
+    if c == 0:
+        assert K.unlinked_unknot_components == 1
+        R = PolynomialRing(ZZ, 't')
+        return R(1)
+    elif c < 100:
         E = Exhaustion(K)
     else:
         E = good_exhaustion(K, max(20, 0.15 * c))

spherogram_src/links/invariants.py

@@ -267,24 +267,43 @@ def alexander_polynomial(self, multivar=True, v='no', method='default',
                 raise RuntimeError('the method "snappy" for '
                                    'alexander_polynomial requires SnapPy')
 
+        # We do any available Type I and II Reidemeister moves as the
+        # functions we call assume that none are available.
+
+        from . import simplify
+        if simplify.has_reidemeister_I_or_II(self):
+            L = self.copy()
+            L.simplify('basic')
+            return L.alexander_polynomial(multivar=multivar, v=v, method=method,
+                                          norm=norm, factored=factored)
+
+        # We have to deal with the special case of unknotted and
+        # unlinked components.
+
         comp = len(self.link_components)
-        if comp < 2:
+        nugatory = self.unlinked_unknot_components
+        if comp + nugatory < 2:
             multivar = False
 
+        if multivar:
+            L = LaurentPolynomialRing(QQ, [f't{i+1}' for i in range(comp + nugatory)])
+            t = list(L.gens())
+        else:
+            L = LaurentPolynomialRing(QQ, 't')
+            t = [L.gen()]
+        R = L.polynomial_ring() if norm else L
+
+        if nugatory > 0:
+            p = 1 if comp + nugatory == 1 else 0
+            return R(p)
+
         # If single variable, use the super-fast method of Bar-Natan.
         if comp == 1 and method == 'default' and norm:
             p = alexander.alexander(self)
         else:  # Use a simple method based on the Wirtinger presentation.
             if method not in ['default', 'wirtinger']:
                 raise ValueError("Available methods are 'default' and 'wirtinger'")
 
-            if multivar:
-                L = LaurentPolynomialRing(QQ, [f't{i+1}' for i in range(comp)])
-                t = list(L.gens())
-            else:
-                L = LaurentPolynomialRing(QQ, 't')
-                t = [L.gen()]
-
             M = self.alexander_matrix(mv=multivar)
             C = M[0]
             m = C.nrows()
@@ -297,7 +316,7 @@ def alexander_polynomial(self, multivar=True, v='no', method='default',
             subMatrix = C[0: k, 0: k]
             p = subMatrix.determinant()
             if p == 0:
-                return 0
+                return R(0)
             if multivar:
                 t_i = M[1][-1]
                 p = (p.factor()) / (t_i - 1)

spherogram_src/links/simplify.py

@@ -140,6 +140,22 @@ def reidemeister_I_and_II(link, A):
     return eliminated, changed
 
 
+def has_reidemeister_I_or_II(link):
+    """
+    Checks if a Reidemeister I or II move is available.
+    """
+    for A in link.crossings:
+        # Type I
+        if A in [B for B, i in A.adjacent]:
+            return True
+        # Type II
+        for a in range(4):
+            (B, b), (C, c) = A.adjacent[a], A.adjacent[a + 1]
+            if B == C and (b - 1) % 4 == c and (a + b) % 2 == 0:
+                return True
+    return False
+        
+
 def basic_simplify(link, build_components=True, to_visit=None,
                    force_build_components=False):
     """

spherogram_src/test.py

@@ -6,13 +6,15 @@
 import spherogram.links.morse
 import spherogram.links.seifert
 import spherogram.links.exhaust
+import spherogram.links.alexander
 import spherogram.links.bridge_bound
 import spherogram.links.bands
 import spherogram.links.bands.merge_links
 import spherogram.links.bands.core
 import spherogram.links.bands.search
 import spherogram.links.bands.regression
 
+
 import spherogram.test_helper as test_helper
 import getopt
 import sys
@@ -31,6 +33,7 @@
            spherogram.links.random_links,
            spherogram.links.orthogonal,
            spherogram.links.simplify,
+           spherogram.links.alexander,
            spherogram.links.bridge_bound,
            spherogram.links.invariants,
            spherogram.links.morse,

spherogram_src/version.py

@@ -1 +1 @@
-version = '2.4'
+version = '2.4.1a1'