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It is known that the SU(2) degrees of freedom manifest in the description ofthe gravitational field in loop quantum gravity are generally reduced to U(1)degrees of freedom on an $S^2$ isolated horizon. General relativity also allowsblack holes with planar, toroidal, or higher genus topology for their horizons.These solutions also meet the criteria for an isolated horizon, save for thetopological criterion, which is not crucial. We discuss the relevantcorresponding symmetry reduction for black holes of various topologies (genus 0and $\geq 2$) here and discuss its ramifications to black hole entropy withinthe loop quantum gravity paradigm. Quantities relevant to the horizon theoryare calculated explicitly using a generalized ansatz for the connection anddensitized triad, as well as utilizing a general metric admitting hyperbolicsub-spaces. In all scenarios, the internal symmetry may be reduced tocombinations of U(1).

A Note on the Symmetry Reduction of SU(2) on Horizons of Various Topologies