Definition of homological longitude

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This paper: https://arxiv.org/abs/1302.6921, mentions 'homological longitude' without giving a definition. Is it the null-homologous longitude in the boundary torus of the knot exterior? This longitude is also referred to as the standard longitude in page 214.

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Yes, the "homological longitude" is the longitude that is nullhomologous in the knot exterior. It's also known as the 0-framing as it has zero linking number with the knot due to it being nullhomologous.