  We introduce a joint generalization, called LRY skein algebras, of Kauffman
bracket skein algebras (of surfaces) that encompasses both Roger-Yang skein
algebras and stated skein algebras. We will show that, over an arbitrary ground
ring which is a commutative domain, the LRY skein algebras are domains and have
degenerations (by filtrations) equal to monomial subalgebras of quantum tori.
For surfaces without interior punctures, this integrality generalizes a result
of Moon and Wong to the most general ground ring. We also calculate the
Gelfand-Kirillov dimension of LRY algebras and show they are Noetherian if the
ground ring is. Moreover they are orderly finitely generated. To study the LRY
algebras and prove the above-mentioned results, we construct quantum traces,
both the so-called X-version for all surfaces and also an A-version for a
smaller class of surfaces. We also introduce a modified version of
Dehn-Thurston coordinates for curves which are more suitable for the study of
skein algebras as they pick up the highest degree terms of products in certain
natural filtrations.
