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My study has led me to (incompressible) branched surfaces as described in https://core.ac.uk/download/pdf/82332579.pdf

In the paper, the authors provide a great example, however I'm looking for more examples not in $S_{g}\times S^{1}$ to help my understanding.

Does anyone have any good (perhaps simple) examples? Maybe some examples of incompressible branched surfaces with boundary?

Thanks in advance!

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Hatcher and Thurston give examples of branched surfaces with boundary in the exteriors of two-bridge knots. These branched surfaces are incompressible whenever each band contains at least two half-twists.

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