2,674 research outputs found
Casson-Lin's invariant of a knot and Floer homology
A. Casson defined an intersection number invariant which can be roughly
thought of as the number of conjugacy classes of irreducible representations of
into counted with signs, where is an oriented integral
homology 3-sphere. X.S. Lin defined an similar invariant (signature of a knot)
to a braid representative of a knot in . In this paper, we give a natural
generalization of the Casson-Lin's invariant to be (instead of using the
instanton Floer homology) the symplectic Floer homology for the representation
space (one singular point) of into with
trace-free along all meridians. The symplectic Floer homology of braids is a
new invariant of knots and its Euler number of such a symplectic Floer homology
is the negative of the Casson-Lin's invariant.Comment: 22 pages, AmsLaTe
Cap-prodcut structures on the Fintushel-Stern spectral sequence
We show that there is a well-defined cap-product structure on the
Fintushel-Stern spectral sequence. Hence we obtain the induced cap-product
structure on the {\BZ}_8-graded instanton Floer homology. The cap-product
structure provides an essentially new property of the instanton Floer homology,
from a topological point of view, which multiplies a finite dimensional
cohomology class by an infinite dimensional homology class (Floer cycles) to
get another infinite dimensional homology class.Comment: 14 pages, no figure, AmsLaTe
A monopole homology of integral homology 3-spheres
To an integral homology 3-sphere , we assign a well-defined -graded
(monopole) homology MH_*(Y, I_{\e}(\T; \e_0)) whose construction in principle
follows from the instanton Floer theory with the dependence of the spectral
flow I_{\e}(\T; \e_0), where \T is the unique U(1)-reducible monopole of
the Seiberg-Witten equation on and \e_0 is a reference perturbation
datum. The definition uses the moduli space of monopoles on Y \x \R
introduced by Seiberg-Witten in studying smooth 4-manifolds. We show that the
monopole homology MH_*(Y, I_{\e}(\T; \e_0)) is invariant among Riemannian
metrics with same I_{\e}(\T; \e_0). This provides a chamber-like structure
for the monopole homology of integral homology 3-spheres. The assigned function
MH_{SWF}: \{I_{\e}(\T; \e_0)\} \to \{MH_*(Y, I_{\e}(\T; \e_0))\} is a
topological invariant (as Seiberg-Witten-Floer Theory).Comment: 20 pages, AMSLaTe
The symplectic Floer homology of the figure eight knot
In this paper, we compute the symplectic Floer homology of the figure eight
knot. This provides first nontrivial knot with trivial symplectic Floer
homology.Comment: LaTeX2e plus AMS style files, 6 page
A Poincar\'e-Hopf type formula for Chern character numbers
For two complex vector bundles admitting a homomorphism with isolated
singularities between them, we establish a Poincar\'e-Hopf type formula for the
difference of the Chern character numbers of these two vector bundles. As a
consequence, we extend the original Poincar\'e-Hopf index formula to the case
of complex vector fields (to appear in Mathematische Zeitschrift)Comment: 10 page
The symplectic Floer homology of composite knots
We develop a method of calculation for the symplectic Floer homology of
composite knots. The symplectic Floer homology of knots defined in \cite{li}
naturally admits an integer graded lifting, and it formulates a filtration and
induced spectral sequence. Such a spectral sequence converges to the symplectic
homology of knots in \cite{li}. We show that there is another spectral sequence
which converges to the -graded symplectic Floer homology for composite
knots represented by braids.Comment: 28pages, AmsLatex, also available at:
http://www.math.okstate.edu/~wli/research/publication.html#recen
Singular connection and Riemann theta function
We prove the Chern-Weil formula for SU(n+1)-singular connections over the
complement of an embedded oriented surface in smooth four manifolds. The
expression of the representation of a number as a sum of nonvanishing squares
is given in terms of the representations of a number as a sum of squares. Using
the number theory result, we study the irreducible SU(n+1)-representations of
the fundamental group of the complement of an embedded oriented surface in
smooth four manifolds.Comment: Latex, 14 page
On the Generalized Volume Conjecture and Regulator
In this paper, by using the regulator map of Beilinson-Deligne on a curve, we
show that the quantization condition posed by Gukov is true for the SL_2(C)
character variety of the hyperbolic knot in S^3. Furthermore, we prove that the
corresponding -valued closed 1-form is a secondary
characteristic class (Chern-Simons) arising from the vanishing first Chern
class of the flat line bundle over the smooth part of the character variety,
where the flat line bundle is the pullback of the universal Heisenberg line
bundle over . Based on this result, we
give a reformulation of Gukov's generalized volume conjecture from a motivic
perspective.Comment: 9 pages, revised version of section 3 of math.GT/0604057, section 3.4
is ne
Massey Product and Twisted Cohomology of A-infinity Algebras
We study the twisted cohomology groups of -algebras defined by
twisting elements and their behavior under morphisms and homotopies using the
bar construction. We define higher Massey products on the cohomology groups of
general -algebras and establish the naturality under morphisms and
their dependency on defining systems. The above constructions are also
considered for -algebras. We construct a spectral sequence converging
to the twisted cohomology groups an show that the higher differentials are
given by the -algebraic Massey products.Comment: 32 page
Volume Conjecture, Regulator and SL_2(C)-Character Variety of a Knot
In this paper, by using the regulator map of Beilinson-Deligne, we show that
the quantization condition posed by Gukov is true for the SL_2(\mathbb{C})
character variety of the hyperbolic knot in S^3. Furthermore, we prove that the
corresponding \mathbb{C}^{*}-valued 1-form is a secondary characteristic class
(Chern-Simons) arising from the vanishing first Chern class of the flat line
bundle over the smooth part of the character variety, where the flat line
bundle is the pullback of the universal Heisenberg line bundle over
\mathbb{C}^{*}\times \mathbb{C}^{*}. The second part of the paper is to define
an algebro-geometric invariant of 3-manifolds resulting from the Dehn surgery
along a hyperbolic knot complement in . We establish a Casson type
invariant for these 3-manifolds. In the last section, we explicitly calculate
the character variety of the figure-eight knot and discuss some applications.Comment: 19 pages, this is the revised and corrected versio
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