In this paper we give a new basis, 螞, for the Homflypt skein mo
dule of the soli
d torus,
d="mmlsi1" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404915001826&_mathId=si1.gif&_user=111111111&_pii=S0022404915001826&_rdoc=1&_issn=00224049&md5=8c8dc9593f366f9002fb0f0c85c4ab9f" title="Click to view the MathML source">S(ST)dden">de">, which topologically is compatible with the han
dle sli
ding moves an
d which was pre
dicte
d by J.H. Przytycki. The basis 螞 is
different from the basis
d="mmlsi2" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404915001826&_mathId=si2.gif&_user=111111111&_pii=S0022404915001826&_rdoc=1&_issn=00224049&md5=b5e12ed4b0e76e7d1c97ae70432c4445" title="Click to view the MathML source">螞′dden">de">,
discovere
d in
depen
dently by Hoste an
d Ki
dwell
d="bbr0010">[1] an
d Turaev
d="bbr0020">[2] with the use of
diagrammatic metho
ds, an
d also
different from the basis of Morton an
d Aiston
d="bbr0030">[3]. For fin
ding the basis 螞 we use the generalize
d Hecke algebra of type B,
d="mmlsi3" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404915001826&_mathId=si3.gif&_user=111111111&_pii=S0022404915001826&_rdoc=1&_issn=00224049&md5=dcda08e1067464280ec360303a1698d5" title="Click to view the MathML source">H1,ndden">de">, which is generate
d by looping elements an
d brai
ding elements an
d which is isomorphic to the affine Hecke algebra of type A
d="bbr0040">[4]. More precisely, we start with the well-known basis
d="mmlsi2" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404915001826&_mathId=si2.gif&_user=111111111&_pii=S0022404915001826&_rdoc=1&_issn=00224049&md5=b5e12ed4b0e76e7d1c97ae70432c4445" title="Click to view the MathML source">螞′dden">de"> of
d="mmlsi1" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404915001826&_mathId=si1.gif&_user=111111111&_pii=S0022404915001826&_rdoc=1&_issn=00224049&md5=8c8dc9593f366f9002fb0f0c85c4ab9f" title="Click to view the MathML source">S(ST)dden">de"> an
d an appropriate linear basis
d="mmlsi322" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404915001826&_mathId=si322.gif&_user=111111111&_pii=S0022404915001826&_rdoc=1&_issn=00224049&md5=5fae6d1e73b15e623783a87675745e22" title="Click to view the MathML source">危ndden">de"> of the algebra
d="mmlsi3" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404915001826&_mathId=si3.gif&_user=111111111&_pii=S0022404915001826&_rdoc=1&_issn=00224049&md5=dcda08e1067464280ec360303a1698d5" title="Click to view the MathML source">H1,ndden">de">. We then convert elements in
d="mmlsi2" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404915001826&_mathId=si2.gif&_user=111111111&_pii=S0022404915001826&_rdoc=1&_issn=00224049&md5=b5e12ed4b0e76e7d1c97ae70432c4445" title="Click to view the MathML source">螞′dden">de"> to sums of elements in
d="mmlsi322" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404915001826&_mathId=si322.gif&_user=111111111&_pii=S0022404915001826&_rdoc=1&_issn=00224049&md5=5fae6d1e73b15e623783a87675745e22" title="Click to view the MathML source">危ndden">de">. Then, using conjugation an
d the stabilization moves, we convert these elements to sums of elements in 螞 by managing gaps in the in
dices, by or
dering the exponents of the looping elements an
d by eliminating brai
ding tails in the wor
ds. Further, we
define total or
derings on the sets
d="mmlsi2" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404915001826&_mathId=si2.gif&_user=111111111&_pii=S0022404915001826&_rdoc=1&_issn=00224049&md5=b5e12ed4b0e76e7d1c97ae70432c4445" title="Click to view the MathML source">螞′dden">de"> an
d 螞 an
d, using these or
derings, we relate the two sets via a block
diagonal matrix, where each block is an infinite lower triangular matrix with invertible elements in the
diagonal. Using this matrix we prove linear in
depen
dence of the set 螞, thus 螞 is a basis for
d="mmlsi1" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404915001826&_mathId=si1.gif&_user=111111111&_pii=S0022404915001826&_rdoc=1&_issn=00224049&md5=8c8dc9593f366f9002fb0f0c85c4ab9f" title="Click to view the MathML source">S(ST)dden">de">.
d="sp0080">d="mmlsi1" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404915001826&_mathId=si1.gif&_user=111111111&_pii=S0022404915001826&_rdoc=1&_issn=00224049&md5=8c8dc9593f366f9002fb0f0c85c4ab9f" title="Click to view the MathML source">S(ST)dden">de"> plays an important role in the study of Homflypt skein modules of arbitrary c.c.o. 3-manifolds, since every c.c.o. 3-manifold can be obtained by integral surgery along a framed link in d="mmlsi5" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404915001826&_mathId=si5.gif&_user=111111111&_pii=S0022404915001826&_rdoc=1&_issn=00224049&md5=9ab3e2ac6abefa020c322a6549af48c7" title="Click to view the MathML source">S3dden">de"> with unknotted components. In particular, the new basis of d="mmlsi1" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404915001826&_mathId=si1.gif&_user=111111111&_pii=S0022404915001826&_rdoc=1&_issn=00224049&md5=8c8dc9593f366f9002fb0f0c85c4ab9f" title="Click to view the MathML source">S(ST)dden">de"> is appropriate for computing the Homflypt skein module of the lens spaces. In this paper we provide some basic algebraic tools for computing skein modules of c.c.o. 3-manifolds via algebraic means.