In thi
s paper we give a new ba
si
s, 螞, for the Homflypt
skein module of the
solid toru
s, <
span id="mml
si1" cla
ss="mathml
src"><
span cla
ss="formulatext
stixSupport mathImg" data-mathURL="/
science?_ob=MathURL&_method=retrieve&_eid=1-
s2.0-S0022404915001826&_mathId=
si1.gif&_u
ser=111111111&_pii=S0022404915001826&_rdoc=1&_i
ssn=00224049&md5=8c8dc9593f366f9002fb0f0c85c4ab9f" title="Click to view the MathML
source">S(ST)
span><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span>, which topologically i
s compatible with the handle
sliding move
s and which wa
s predicted by J.H. Przytycki. The ba
si
s 螞 i
s different from the ba
si
s <
span id="mml
si2" cla
ss="mathml
src"><
span cla
ss="formulatext
stixSupport mathImg" data-mathURL="/
science?_ob=MathURL&_method=retrieve&_eid=1-
s2.0-S0022404915001826&_mathId=
si2.gif&_u
ser=111111111&_pii=S0022404915001826&_rdoc=1&_i
ssn=00224049&md5=b5e12ed4b0e76e7d1c97ae70432c4445" title="Click to view the MathML
source">螞<
sup>′
sup>
span><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span>, di
scovered independently by Ho
ste and Kidwell <
span id="bbr0010">[1]
span> and Turaev <
span id="bbr0020">[2]
span> with the u
se of diagrammatic method
s, and al
so different from the ba
si
s of Morton and Ai
ston <
span id="bbr0030">[3]
span>. For finding the ba
si
s 螞 we u
se the generalized Hecke algebra of type B, <
span id="mml
si3" cla
ss="mathml
src"><
span cla
ss="formulatext
stixSupport mathImg" data-mathURL="/
science?_ob=MathURL&_method=retrieve&_eid=1-
s2.0-S0022404915001826&_mathId=
si3.gif&_u
ser=111111111&_pii=S0022404915001826&_rdoc=1&_i
ssn=00224049&md5=dcda08e1067464280ec360303a1698d5" title="Click to view the MathML
source">H<
sub>1,n
sub>
span><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span>, which i
s generated by looping element
s and braiding element
s and which i
s i
somorphic to the affine Hecke algebra of type A <
span id="bbr0040">[4]
span>. More preci
sely, we
start with the well-known ba
si
s <
span id="mml
si2" cla
ss="mathml
src"><
span cla
ss="formulatext
stixSupport mathImg" data-mathURL="/
science?_ob=MathURL&_method=retrieve&_eid=1-
s2.0-S0022404915001826&_mathId=
si2.gif&_u
ser=111111111&_pii=S0022404915001826&_rdoc=1&_i
ssn=00224049&md5=b5e12ed4b0e76e7d1c97ae70432c4445" title="Click to view the MathML
source">螞<
sup>′
sup>
span><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span> of <
span id="mml
si1" cla
ss="mathml
src"><
span cla
ss="formulatext
stixSupport mathImg" data-mathURL="/
science?_ob=MathURL&_method=retrieve&_eid=1-
s2.0-S0022404915001826&_mathId=
si1.gif&_u
ser=111111111&_pii=S0022404915001826&_rdoc=1&_i
ssn=00224049&md5=8c8dc9593f366f9002fb0f0c85c4ab9f" title="Click to view the MathML
source">S(ST)
span><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span> and an appropriate linear ba
si
s <
span id="mml
si322" cla
ss="mathml
src"><
span cla
ss="formulatext
stixSupport mathImg" data-mathURL="/
science?_ob=MathURL&_method=retrieve&_eid=1-
s2.0-S0022404915001826&_mathId=
si322.gif&_u
ser=111111111&_pii=S0022404915001826&_rdoc=1&_i
ssn=00224049&md5=5fae6d1e73b15e623783a87675745e22" title="Click to view the MathML
source">危<
sub>n
sub>
span><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span> of the algebra <
span id="mml
si3" cla
ss="mathml
src"><
span cla
ss="formulatext
stixSupport mathImg" data-mathURL="/
science?_ob=MathURL&_method=retrieve&_eid=1-
s2.0-S0022404915001826&_mathId=
si3.gif&_u
ser=111111111&_pii=S0022404915001826&_rdoc=1&_i
ssn=00224049&md5=dcda08e1067464280ec360303a1698d5" title="Click to view the MathML
source">H<
sub>1,n
sub>
span><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span>. We then convert element
s in <
span id="mml
si2" cla
ss="mathml
src"><
span cla
ss="formulatext
stixSupport mathImg" data-mathURL="/
science?_ob=MathURL&_method=retrieve&_eid=1-
s2.0-S0022404915001826&_mathId=
si2.gif&_u
ser=111111111&_pii=S0022404915001826&_rdoc=1&_i
ssn=00224049&md5=b5e12ed4b0e76e7d1c97ae70432c4445" title="Click to view the MathML
source">螞<
sup>′
sup>
span><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span> to
sum
s of element
s in <
span id="mml
si322" cla
ss="mathml
src"><
span cla
ss="formulatext
stixSupport mathImg" data-mathURL="/
science?_ob=MathURL&_method=retrieve&_eid=1-
s2.0-S0022404915001826&_mathId=
si322.gif&_u
ser=111111111&_pii=S0022404915001826&_rdoc=1&_i
ssn=00224049&md5=5fae6d1e73b15e623783a87675745e22" title="Click to view the MathML
source">危<
sub>n
sub>
span><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span>. Then, u
sing conjugation and the
stabilization move
s, we convert the
se element
s to
sum
s of element
s in 螞 by managing gap
s in the indice
s, by ordering the exponent
s of the looping element
s and by eliminating braiding tail
s in the word
s. Further, we define total ordering
s on the
set
s <
span id="mml
si2" cla
ss="mathml
src"><
span cla
ss="formulatext
stixSupport mathImg" data-mathURL="/
science?_ob=MathURL&_method=retrieve&_eid=1-
s2.0-S0022404915001826&_mathId=
si2.gif&_u
ser=111111111&_pii=S0022404915001826&_rdoc=1&_i
ssn=00224049&md5=b5e12ed4b0e76e7d1c97ae70432c4445" title="Click to view the MathML
source">螞<
sup>′
sup>
span><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span> and 螞 and, u
sing the
se ordering
s, we relate the two
set
s via a block diagonal matrix, where each block i
s an infinite lower triangular matrix with invertible element
s in the diagonal. U
sing thi
s matrix we prove linear independence of the
set 螞, thu
s 螞 i
s a ba
si
s for <
span id="mml
si1" cla
ss="mathml
src"><
span cla
ss="formulatext
stixSupport mathImg" data-mathURL="/
science?_ob=MathURL&_method=retrieve&_eid=1-
s2.0-S0022404915001826&_mathId=
si1.gif&_u
ser=111111111&_pii=S0022404915001826&_rdoc=1&_i
ssn=00224049&md5=8c8dc9593f366f9002fb0f0c85c4ab9f" title="Click to view the MathML
source">S(ST)
span><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span>.
sp0080"><span id="mmlsi1" class="mathmlsrc"><span class="formulatext stixSupport mathImg" 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)span><span class="mathContainer hidden"><span class="mathCode">span>span>span> 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 <span id="mmlsi5" class="mathmlsrc"><span class="formulatext stixSupport mathImg" 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">S<sup>3sup>span><span class="mathContainer hidden"><span class="mathCode">span>span>span> with unknotted components. In particular, the new basis of <span id="mmlsi1" class="mathmlsrc"><span class="formulatext stixSupport mathImg" 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)span><span class="mathContainer hidden"><span class="mathCode">span>span>span> 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.