  We introduce two new bases of the ring of polynomials and study their
relations to known bases. The first basis is the quasiLascoux basis, which is
simultaneously both a $K$-theoretic deformation of the quasikey basis and also
a lift of the $K$-analogue of the quasiSchur basis from quasisymmetric
polynomials to general polynomials. We give positive expansions of this
quasiLascoux basis into the glide and Lascoux atom bases, as well as a positive
expansion of the Lascoux basis into the quasiLascoux basis. As a special case,
these expansions give the first proof that the $K$-analogues of quasiSchur
polynomials expand positively in multifundamental quasisymmetric polynomials of
T. Lam and P. Pylyavskyy.
  The second new basis is the kaon basis, a $K$-theoretic deformation of the
fundamental particle basis. We give positive expansions of the glide and
Lascoux atom bases into this kaon basis.
  Throughout, we explore how the relationships among these $K$-analogues mirror
the relationships among their cohomological counterparts. We make several
'alternating sum' conjectures that are suggestive of Euler characteristic
calculations.
