doi:10.1016/j.jalgebra.2016.11.013engBenkart, G.Elduque, A.Cross products, invariants, and centralizersART-2018-105220An 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.2018http://zaguan.unizar.es/record/7069810.1016/j.jalgebra.2016.11.013http://zaguan.unizar.es/record/70698oai:zaguan.unizar.es:70698info:eu-repo/grantAgreement/ES/DGA/FSEinfo:eu-repo/grantAgreement/ES/MINECO/MTM2013-45588-C3-2-PJOURNAL OF ALGEBRA 500 (2018), 69-102by-nc-ndhttp://creativecommons.org/licenses/by-nc-nd/3.0/es/info:eu-repo/semantics/openAccess