  We define a type of Niebrzydowski tribracket we call $\Delta$-tribrackets and
show that their counting invariants are invariants of link-homotopy. We further
identify several classes of tribrackets whose counting invariants for oriented
classical knots and links are trivial, including vertical tribrackets
satisfying the center-involutory condition and horizontal tribrackets
satisfying the late-commutativity condition. We provide examples and end with
questions for future research.
