$\mathrm{\Delta }$-Tribrackets and Link Homotopy

We define a type of Niebrzydowski tribracket we call $\mathrm{\Delta }$-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.