Resumen: An algebra V with a cross product x has dimension 3 or 7. In this work, we use 3-tangles to describe, and provide a basis for, the space of homomorphisms from V-circle times n to V-circle times m that are invariant under the action of the automorphism group Aut(V, x) of V, which is a special orthogonal group when dim V = 3, and a simple algebraic group of type G(2) when dim V = 7. When m = n, this gives a graphical description of the centralizer algebra End(Aut(v, x))(V-circle times n), and therefore, also a graphical realization of the Aut(V, x)-invariants in V-circle times 2n equivalent to the First Fundamental Theorem of Invariant Theory. We show how the 3-dimensional simple Kaplansky Jordan superalgebra can be interpreted as a cross product (super)algebra and use 3-tangles to obtain a graphical description of the centralizers and invariants of the Kaplansky superalgebra relative to the action of the special orthosymplectic group. Idioma: Inglés DOI: 10.1016/j.jalgebra.2016.11.013 Año: 2018 Publicado en: JOURNAL OF ALGEBRA 500 (2018), 69-102 ISSN: 0021-8693 Factor impacto JCR: 0.666 (2018) Categ. JCR: MATHEMATICS rank: 189 / 313 = 0.604 (2018) - Q3 - T2 Factor impacto SCIMAGO: 1.137 - Algebra and Number Theory (Q1)