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 $\pi_1(Y)$ into $SU(2)$ counted with signs, where $Y$ 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 $S^3$. 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 $\pi_1(S^3 \setminus K)$ into $SU(2)$ 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 $Y$, we assign a well-defined $\Z$-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 $Y$ 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 $\Z$-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 $\mathbb{C}^{*}$-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 $\mathbb{C}^{*}\times \mathbb{C}^{*}$. 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 $A_\infty$-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 $A_\infty$-algebras and establish the naturality under morphisms and their dependency on defining systems. The above constructions are also considered for $C_\infty$-algebras. We construct a spectral sequence converging to the twisted cohomology groups an show that the higher differentials are given by the $A_\infty$-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 $S^3$. 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