Flipping relation as a reduced star-star relation

Abstract In this paper, we consider the lens hyperbolic gamma solution to the star-star relation and the flipping relation from three-dimensional $$\mathcal {N}=2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> supersymmetric gauge theories on $$S^3_b/\mathbb {Z}_r$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>b</mml:mi> <mml:mn>3</mml:mn> </mml:msubsup> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>Z</mml:mi> <mml:mi>r</mml:mi> </mml:msub> </mml:mrow> </mml:math> . We explore that a certain limit of the star-star relation yields the latter symmetry transformation, which exchanges the edge interactions of two outer spins with a centrally sited spin. Furthermore, we obtain more solutions to the flipping relation in terms of the hyperbolic gamma, basic hypergeometric, and the Euler gamma functions.