There is also a general method for computing inverses in a finite algebra, which calling super().__invert__() should go to. (Although it is computationally intensive.)

I will take that into account. Once we have integrated representation theory (via the cellular algebra structure) into the class, it will make sense to add this as a fallback option. Currently, however, we do not have a working to_matrix method:

File ~/devel/sage/src/sage/categories/finite_dimensional_algebras_with_basis.py:1389, in FiniteDimensionalAlgebrasWithBasis.ElementMethods.__invert__(self)
   1386             raise ValueError("cannot invert self (= %s)" % self)
   1388 e = alg.one().to_vector()
-> 1389 A = self.to_matrix()
...
File ~/devel/sage/src/sage/monoids/tangles.py:898, in KauffmanTangles.morton_wasserman_tangle(self, bd, top_bottom)
    895 from sage.misc.flatten import flatten
    897 n = self.strands()
--> 898 nb = max(bd.base_set())
    899 if n != nb:
    900     raise ValueError('Base set of Brauer diagram is incompatible with the number of strands %s' % n)

File sage/structure/element.pyx:494, in sage.structure.element.Element.__getattr__()

File sage/structure/element.pyx:507, in sage.structure.element.Element.getattr_from_category()

File sage/cpython/getattr.pyx:357, in sage.cpython.getattr.getattr_from_other_class()

AttributeError: 'KauffmanTangles_with_category.element_class' object has no attribute 'base_set'

The to_matrix() method should be returning the result of the algebra acting on itself (i.e., as the regular representation). So this should work because multiplication of the algebra on itself must be well-defined (and you don't seem to override it). This seems like it's exposing a genuine bug.

The to_matrix() method should be returning the result of the algebra acting on itself (i.e., as the regular representation). So this should work because multiplication of the algebra on itself must be well-defined (and you don't seem to override it). This seems like it's exposing a genuine bug.

The cause likely lies in my specific basic construction. This is based on an iterated, lazy family that, in a first step, maps the Brauer diagrams to Morton-Wassermann tangles and then the tangles to the monomials of the BMW algebra:

       brauer_basis = self._brauer_algebra.basis().keys()
       basis = Family(brauer_basis, function=KT.morton_wasserman_tangle, lazy=True, name='Morton-Wasserman-Tangle')

sage: algebras.BirmanMurakamiWenzl(3).basis()
Lazy family (Term map from Lazy family (Morton-Wasserman-Tangle(i))_{i in Brauer diagrams of order 3} to Birman-Murakami-Wenzl algebra on 3 strands over Multivariate Laurent Polynomial Ring in l, m over Integer Ring(i))_{i in Lazy family (Cached version of <function KauffmanTangles.morton_wasserman_tangle at 0x7faefbc87420>(i))_{i in Brauer diagrams of order 3}}

Perhaps this is a bit too complicated. My intention was to display the basis elements as tangles, not Brauer diagrams. I would appreciate it if you could suggest a better solution.

I think it would be better to store this in the parent as a @lazy_attribute.

Done!

Do you also want a _latex_ method? Not necessary though.

It is difficult to find a consistent convention in the literature. In the reference by Morton and Wasserman, the set of n-m tangles is denoted as {\cal U}_n^m. Elsewhere, the monoidal category of tangles is denoted as {\cal T}, so these monoids would be denoted as Hom_{\cal T}(n, n). Which convention would you prefer?

but perhaps better at the class level docstring?

Done!