diff --git a/.github/workflows/build.yml b/.github/workflows/build.yml
index 824c015..7dfc777 100644
--- a/.github/workflows/build.yml
+++ b/.github/workflows/build.yml
@@ -8,13 +8,13 @@ jobs:
     runs-on: ${{ matrix.os }}
     strategy:
       matrix:
-        os: [ubuntu-latest, macos-13, macos-14, windows-2022]
+        os: [ubuntu-latest, ubuntu-latest-arm, macos-15-intel, macos-latest, windows-2022]
 
     steps:
       - uses: actions/checkout@v4
 
       - name: Build wheels
-        uses: pypa/cibuildwheel@v2.22
+        uses: pypa/cibuildwheel@v3.3
 
       - uses: actions/upload-artifact@v4
         with:
diff --git a/.github/workflows/test.yml b/.github/workflows/test.yml
index 11ae3a9..b50e223 100644
--- a/.github/workflows/test.yml
+++ b/.github/workflows/test.yml
@@ -9,7 +9,7 @@ jobs:
     runs-on: ubuntu-latest
     strategy:
       matrix:
-        version: [9.3, 9.4, 9.5, 9.6, 9.7, 9.8, 10.1, 10.2, 10.3, 10.4, latest]
+        version: [9.3, 9.4, 9.5, 9.6, 9.7, 9.8, 10.1, 10.2, 10.3, 10.4, 10.5, 10.6, 10.7, latest]
       fail-fast: false
 
     container:
diff --git a/pyproject.toml b/pyproject.toml
index 04614f5..10de76f 100644
--- a/pyproject.toml
+++ b/pyproject.toml
@@ -3,8 +3,8 @@ requires = ["setuptools", "wheel", "cython"]
 build-backend = "setuptools.build_meta"
 
 [tool.cibuildwheel]
-build = "cp38* cp39* cp310* cp311* cp312* cp313*"
-skip = "*musllinux*"
+build = "cp38* cp39* cp310* cp311* cp312* cp313* cp314*"
+skip = "*musllinux* cp313t* cp314t*"
 test-command = "python -m spherogram.test"
 
 [tool.cibuildwheel.macos]
diff --git a/spherogram_src/links/bands/merge_links.py b/spherogram_src/links/bands/merge_links.py
index bde6691..e9995b7 100644
--- a/spherogram_src/links/bands/merge_links.py
+++ b/spherogram_src/links/bands/merge_links.py
@@ -33,6 +33,17 @@ def are_isometric_as_links(A, B):
         return False
 
 
+def are_same_triangulations_and_links(A, B):
+    if hasattr(A, 'exterior'):
+        A = A.exterior(with_hyperbolic_structure=False)
+
+    if hasattr(B, 'exterior'):
+        B = B.exterior(with_hyperbolic_structure=False)
+
+    isos = A.isomorphisms_to(B)
+    return any(iso.extends_to_link() for iso in isos)
+
+
 def link_isotopy_classes(links_with_manifolds):
     """
     Given a collection of links, try to select one from each isotopy
diff --git a/spherogram_src/links/bands/search.py b/spherogram_src/links/bands/search.py
index b49e68d..ad48326 100644
--- a/spherogram_src/links/bands/search.py
+++ b/spherogram_src/links/bands/search.py
@@ -1,5 +1,7 @@
 from ..links import Link
-from .merge_links import link_isotopy_classes, are_isometric_as_links
+from .merge_links import (link_isotopy_classes,
+                          are_isometric_as_links,
+                          are_same_triangulations_and_links)
 from .core import Band, add_one_band, banded_links, normalize_crossing_labels
 
 
@@ -81,6 +83,33 @@ def remove_reidemeister_I(link):
         link._build_components(component_starts)
     return success
 
+def are_same_link(L0, L1, tries=10):
+    """
+    Try to determine whether the given links are isotopic *or* one is
+    isotopic to the mirror of the other.
+
+    If it returns True, then the links are definitely equivalent.  If
+    it returns False, then they may still be the same.
+
+    When both links are hyperbolic, this is function is robust.
+    Otherwise, whether it succeeds is pretty stochastic.
+    """
+    L0, L1 = L0.copy(), L1.copy()
+    for i in range(tries):
+        if (L0.PD_code() == L1.PD_code() or
+            are_isometric_as_links(L0, L1) or
+            are_same_triangulations_and_links(L0, L1)):
+            return True
+
+        L0.backtrack(30)
+        L0.simplify()
+        L0.simplify('global')
+        L1.backtrack(30)
+        L1.simplify()
+        L1.simplify('global')
+
+    return False
+
 
 def verify_ribbon_to_unknot(link, certificate):
     """
@@ -92,22 +121,29 @@ def verify_ribbon_to_unknot(link, certificate):
     >>> verify_ribbon_to_unknot(L, cert)  #doctest: +SNAPPY
     True
 
-    TODO: Doesn't work when some intermediate link is non-hyperbolic,
-    in this case a connected sum of two trefoils.  The below should
-    return True
+    TODO: Sometimes doesn't work when some intermediate link is
+    non-hyperbolic.  in this case a connected sum of two trefoils.
+    The below should return True but might fail occasionally.
 
     >>> L = Link([(0,28,1,27),(10,1,11,2),(2,11,3,12),(36,3,37,4),(19,5,20,4),(5,23,6,22),(15,6,16,7),(34,8,35,7),(8,25,9,26),(26,9,27,10),(12,36,13,35),(13,20,14,21),(21,14,22,15),(23,17,24,16),(17,31,18,30),(31,19,32,18),(24,33,25,34),(28,0,29,37),(29,33,30,32)])
     >>> cert = [[(0,28,1,27),(10,1,11,2),(2,11,3,12),(36,3,37,4),(19,5,20,4),(5,23,6,22),(15,6,16,7),(34,8,35,7),(8,25,9,26),(26,9,27,10),(12,36,13,35),(13,20,14,21),(21,14,22,15),(23,17,24,16),(17,31,18,30),(31,19,32,18),(24,33,25,34),(28,0,29,37),(29,33,30,32)],'483d_0_0',[(5,2,6,3),(3,6,4,7),(1,4,2,5),(7,11,8,10),(11,9,0,8),(9,1,10,0)],'1603_0_1','unknot']
     >>> verify_ribbon_to_unknot(L, cert)  #doctest: +SNAPPY
-    False
+    True
     """
-    assert certificate[-1] == 'unknot'
+    import snappy
+    if certificate[-1] != 'unknot':
+        try:
+            snappy.RibbonLinks[certificate[-1]]
+        except KeyError:
+            raise ValueError('ribbon_cert does not end with known ribbon link')
+
     L = link.copy()
     E = L.exterior()
     for i in range(0, len(certificate) - 1, 2):
         L_cert = Link(certificate[i])
         E_cert = L_cert.exterior()
-        if not (L.PD_code() == L_cert.PD_code() or are_isometric_as_links(E, E_cert)):
+
+        if not are_same_link(L, L_cert):
             return False
 
         band = Band(certificate[i + 1])
@@ -123,6 +159,10 @@ def verify_ribbon_to_unknot(link, certificate):
         L.unlinked_unknot_components = 0
         E = L.exterior()
 
+    if certificate[-1] != 'unknot':
+        E_cert = snappy.RibbonLinks[certificate[-1]]
+        return are_isometric_as_links(E, E_cert)
+
     return len(L.link_components) == 0 or is_unlink_exterior(E)