A homology cylinder of a surface induces an automorphism of the completed
group ring of the fundamental group of the surface. We introduce a new method
of computing the automorphism by using the Goldman Lie algebra of the surface
or some skein algebra. In particular, we give a refinement of a formula by Kuno
and Massuyeau.

arXiv

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Other Link： http://arxiv.org/pdf/2010.03764v1

The generalized Dehn twist along a closed curve in an oriented surface is an
algebraic construction which involves intersections of loops in the surface. It
is defined as an automorphism of the Malcev completion of the fundamental group
of the surface. As the name suggests, for the case where the curve has no
self-intersection, it is induced from the usual Dehn twist along the curve. In
this expository article, after explaining their definition, we review several
results about generalized Dehn twists such as their realizability as
diffeomorphisms of the surface, their diagrammatic description in terms of
decorated trees and the Hopf-algebraic framework underlying their construction.
Going to the dimension three, we also overview the relation between generalized
Dehn twists and $3$-dimensional homology cobordisms, and we survey the variants
of generalized Dehn twists for skein algebras of the surface.

arXiv

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Other Link： http://arxiv.org/pdf/1909.09496v1

We introduce a formula for the action of Dehn twists on the HOMFLY-PT type
skein module of a surface. As an application of the formula to mapping class
group, we give an embedding from the Torelli group of a surface $\Sigma_{g,1}$
of genus $g$ with non-empty connected boundary into the completed HOMFLY-PT
type skein algebra. As an application of the formula to integral homology
$3$-spheres, we construct an invariant $z(M) \in \mathbb{Q} [\rho ] [[h]]$ for
an integral homology $3$-sphere $M$. The invariant $z(M) \mod (h^{n+1})$ is a
finite type invariant of order $n$.

arXiv

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Other Link： http://arxiv.org/pdf/1801.00580v1

We construct an invariant $z (M) =1+a_1(A^4-1)+ a_2(A^4-1)^2+a_3(A^4-1)^3 +
\cdots \in \mathbb{Q} [[A^4-1]]= \mathbb{Q} [[A+1]]$ for an integral homology
$3$-sphere $M$ using a completed skein algebra and a Heegaard splitting. The
invariant $z(M)\mathrm{mod} ((A+1)^{n+1}) $ is a finite type invariant of order
$n$. In particular, $-a_1/6$ equals the Casson invariant. If $M$ is the
Poincar\'{e} homology 3-sphere, $(z(M))_{|A^4 =q} \mod (q+1)^{14} $ is the
Ohtsuki series for $M$.

arXiv

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Other Link： http://arxiv.org/pdf/1607.01580v4

Language：English  

DOI： 10.1017/S0305004117000366

Web of Science

arXiv

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Other Link： http://arxiv.org/pdf/1606.01114v2