Rotating and potentially inverting word to simplify.

python/len_spec/__init__.py

@@ -2,7 +2,7 @@
 from .tile import compute_length_spectrum_tiles, LengthSpectrumTile
 from .geodesic_info import GeodesicInfoBase, GeodesicKeyInfo, CoreCurveGeodesicInfo
 from .geodesic_key_info_dict import get_geodesic_key_info_set
-from .word import simplify_cyclic_word
+from .word import simplify_geodesic_word
 from .length_spectrum_geodesic_info import LengthSpectrumGeodesicInfo
 
 from ..geometric_structure import word_list_to_psl2c_matrix
@@ -31,13 +31,19 @@ def length_spectrum(manifold,
         >>> spec = M._length_spectrum_dev()
         >>> next(spec) # doctest: +NUMERIC9
         Length                                       Word          Core curve
-        0.14742465268510 - 1.78287093565206*I        CBAdCBAAb     Cusp 0
+        0.14742465268510 - 1.78287093565206*I        aabcDabcB     Cusp 0
         >>> next(spec) # doctest: +NUMERIC9
-        0.81161414965959 + 2.72911699294425*I        B
+        0.81161414965959 + 2.72911699294425*I        b             -
         >>> next(spec) # doctest: +NUMERIC9
         0.84163270359334 + 2.61245944742151*I        aB            -
         >>> next(spec) # doctest: +NUMERIC9
-        0.93461379591349 + 2.70060614107722*I        A             -
+        0.93461379591349 + 2.70060614107722*I        a             -
+        >>> next(spec) # doctest: +NUMERIC9
+        1.50642474995261 + 3.13101560181284*I        ab            -
+        >>> next(spec) # doctest: +NUMERIC9
+        1.58977016368115 - 3.08892331549756*I        abC           -
+        >>> next(spec) # doctest: +NUMERIC9
+        1.60467656123231 + 3.09975771482376*I        aBB           -
 
         >>> M = Manifold("m125(3,4)(0,0)")
         >>> spec = M._length_spectrum_dev()
@@ -74,7 +80,7 @@ def length_spectrum(manifold,
             _is_first = is_first,
             length = geodesic.length,
             word = list_as_word(
-                simplify_cyclic_word(geodesic.word),
+                simplify_geodesic_word(geodesic.word),
                 num_generators,
                 verbose_form=False),
             core_curve = core_curve,

python/len_spec/word.py

@@ -1,27 +1,30 @@
-from ..SnapPy import reduce_list_word
+from ..SnapPy import reduce_list_word, inverse_list_word
 
 from typing import List
 
-def simplify_cyclic_word(word : List[int]) -> List[int]:
+def simplify_geodesic_word(word : List[int]) -> List[int]:
     """
     Cancels generator and inverse generator - whether they are next to each
     other or at opposite ends of the word.
 
-    >>> simplify_cyclic_word([])
+    >>> simplify_geodesic_word([])
     []
-    >>> simplify_cyclic_word([1])
+    >>> simplify_geodesic_word([1])
     [1]
-    >>> simplify_cyclic_word([1, 2])
+    >>> simplify_geodesic_word([1, 2])
     [1, 2]
-    >>> simplify_cyclic_word([-1, 2, -2, 1])
+    >>> simplify_geodesic_word([-1, 2, -2, 1])
     []
-    >>> simplify_cyclic_word([-1, 3, 2, -2, 1])
+    >>> simplify_geodesic_word([-1, 3, 2, -2, 1])
     [3]
-    >>> simplify_cyclic_word([-1, 3, 4, 2, -2, 1])
+    >>> simplify_geodesic_word([-1, 3, 4, 2, -2, 1])
     [3, 4]
     """
 
-    return _cancel_conjugation(reduce_list_word(word))
+    return (
+        _rotate_and_optionally_invert(
+            _cancel_conjugation(
+                reduce_list_word(word))))
 
 def _cancel_conjugation(word : List[int]) -> List[int]:
     n = len(word)
@@ -30,3 +33,17 @@ def _cancel_conjugation(word : List[int]) -> List[int]:
         if word[i] != -word[k]:
             return word[i:k+1]
     return word[n // 2 : (n + 1) // 2]
+
+def _rotate_and_optionally_invert(word : List[int]) -> List[int]:
+    n = len(word)
+
+    if n == 0:
+        return word
+
+    return min(
+        (candidates[i:] + candidates[:i]
+         for candidates in [ word, inverse_list_word(word) ]
+         for i in range(len(word))),
+        key=lambda w: [ 2 * l if l > 0 else 2 * -l + 1
+                        for l in w ])
+