Michael Polyak, Oleg Viro
We study the Vassiliev knot invariant v_2 of degree 2. We present it via the degrees of maps of various configuration spaces related to a knot to products of spheres. This gives rise to numerous geometrical and combinatorial formulas for this invariant.
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I present a formula for the Casson invariant of knots associated with divides. The formula is written in terms of Arnold's invariants of pieces of the divide. Various corollaries are discussed.
It is known that if any prime power branched cyclic cover of a knot in the 3-sphere is a homology sphere, then the knot has vanishing Casson-Gordon invariants. We construct infinitely many examples of (topologically) non-slice knots in the 3-sphere whose prime power branched cyclic covers are homology spheres. We show that these knots generate an infinite rank subgroup of F_(1.0)/F_(1.5) for which Casson-Gordon invariants vanish in Cochran-Orr-Teichner's filtration of the classical knot concordance group . As a corollary, it follows that Casson-Gordon invariants are not a complete set of obstr…
Cosmetic surgeries on knots in homology spheres and the Casson--Walker invariant
针对同调球中纽结的美容手术问题，本文利用三维球内2分支链环的 Casson–Walker 不变量有理手术公式，给出纽结与手术斜率存在此类手术时的若干约束。
In this paper, we extend the definition of the $SL_2(\Bbb C)$ Casson invariant to arbitrary knots $K$ in integral homology 3-spheres and relate it to the $m$-degree of the $\widehat{A}$-polynomial of $K$. We prove a product formula for the $\widehat{A}$-polynomial of the connected sum $K_1 \# K_2$ of two knots in $S^3$ and deduce additivity of $SL_2(\Bbb C)$ Casson knot invariant under connected sum for a large class of knots in $S^3$. We also present an example of a nontrivial knot $K$ in $S^3$ with trivial $\widehat{A}$-polynomial and trivial $SL_2(\Bbb C)$ Casson knot invariant, showing tha…
For a knot $K$ with $Δ_K(t)\doteq t^2-3t+1$ in a homology $3$-sphere, let $M$ be the result of $2/q$-surgery on $K$. We show that appropriate assumptions on the Reidemeister torsion and the Casson-Walker-Lescop invariant of the universal abelian covering of $M$ imply $q=\pm 1$, if $M$ is a Seifert fibered space.
We derive a simple closed formula for the SL(2,C) Casson invariant for Seifert fibered homology 3-spheres using the correspondence between SL(2,C) character varieties and moduli spaces of parabolic Higgs bundles of rank two. These results are then used to deduce the invariant for Dehn surgeries on twist knots by combining computations of the Culler-Shalen norms with the surgery formula for the SL(2,C) Casson invariant.
We give a rational surgery formula for the Casson-Walker invariant of a 2-component link in $S^{3}$ which is a generalization of Matveev-Polyak's formula. As application, we give more examples of non-hyperbolic L-space $M$ such that knots in $M$ are determined by their complements. We also apply the result for the cosmetic crossing conjecture.
In this paper we prove that the Casson-Gordon invariants of the connected sum of two knots split when the Alexander polynomials of the knots are coprime. As one application, for any knot K, all but finitely many algebraically slice twisted doubles of K are linearly independent in the knot concordance group.
For a knot $K$ with $Δ_K(t)\doteq t^2-3t+1$ in a homology $3$-sphere, let $M$ be the result of $2/q$-surgery on $K$. We show that an appropriate assumption on the Reidemeister torsion of the universal abelian covering of $M$ implies $q=\pm 1$, if $M$ is a Seifert fibered space.
We consider the cosmetic surgery problem for two-bridge knots in the 3-sphere. It is seen that all the two-bridge knots at most 9 crossings other than $9_{27} = S(49,19)=C[2,2,-2,2,2,-2]$ admits no purely cosmetic surgery pairs. Then we show that any two-bridge knot of the Conway form $[2x,2,-2x,2x,2,-2x]$ with $x \ge 1$ admits no cosmetic surgery pairs yielding homology 3-spheres, where $9_{27}$ appears for $x=1$. Our advantage to prove this is using the $SL(2,\mathbb{C})$ Casson invariant.
We investigate the behavior of the SL(2,C) Casson invariant for 3-manifolds obtained by Dehn surgery along two-bridge knots. Using the results of Hatcher and Thurston, and also results of Ohtsuki, we outline how to compute the Culler--Shalen seminorms, and we illustrate this approach by providing explicit computations for double twist knots. We then apply the surgery formula of Curtis to deduce the SL(2,C) Casson invariant for the 3-manifolds obtained by p/q-Dehn surgery on such knots. These results are applied to prove nontriviality of the SL(2,C) Casson invariant for nearly all 3-manifolds o…
We consider an analogue of well-known Casson knot invariant for knotoids. We start with a direct analogue of the classical construction which gives two different integer-valued knotoid invariants and then focus on its homology extension. Value of the extension is a formal sum of subgroups of the first homology group $H_1(Σ)$ where $Σ$ is an oriented surface with (maybe) non-empty boundary in which knotoid diagrams lie. To make the extension informative for spherical knotoids it is sufficient to transform an initial knotoid diagram in $S^2$ into a knotoid diagram in the annulus by removing smal…
Let K and K' be 2-knots. Suppose that K and K' are ribbon-move equivalent. Then the Farber-Levine pairing for K is equivalent to that for K' and the (Z-)torsion part of the first Alexander module of $K$ is isomorphic to that of K' as Z[Z] modules. Let K be a 2-knot which is ribbon-move equivalent to the trivial knot. Then the Atiyah-Patodi-Singer-Casson-Gordon-Ruberman Q/Z-valued \widetildeη-invariants of K for Z_d is zero. (d is a natural number. d>2.).
We show that the SU(3) Casson invariant for spliced sums along certain torus knots equals 16 times the product of their SU(2) Casson knot invariants. The key step is a splitting formula for su(n) spectral flow for closed 3-manifolds split along a torus.
We establish a formula for the SL(2,C) Casson invariant of spliced sums of homology spheres along knots. Along the way, we show that the SL(2,C) Casson invariant vanishes for spliced sums along knots in the 3-sphere.
In the study of homology cobordisms, knot concordance and link concordance, the following technical problem arises frequently: let $π$ be a group and let $M \to N$ be a homomorphism between projective $\Z[π]$-modules such that $\Z_p \otimes_{\Z[π]} M\to \Z_p \otimes_{\Z[π]} N$ is injective; for which other right $\Z[π]$-modules $V$ is the induced map $V \otimes_{\Z[π]} M\to V\otimes_{\Z[π]}N$ also injective? Our main theorem gives a new criterion which combines and generalizes many previous results.
For a knot $K$ in a homology $3$-sphere $Σ$, let $M$ be the result of $2/q$-surgery on $K$, and let $X$ be the universal abelian covering of $M$. Our first theorem is that if the first homology of $X$ is finite cyclic and $M$ is a Seifert fibered space with $N\ge 3$ singular fibers, then $N\ge 4$ if and only if the first homology of the universal abelian covering of $X$ is infinite. Our second theorem is that under an appropriate assumption on the Alexander polynomial of $K$, if $M$ is a Seifert fibered space, then $q=\pm 1$ (i.e.\ integral surgery).
The n-solvable filtration $\{\mathcal{F}_n\}_{n=0}^\infty$ of the smooth knot concordance group (denoted by $\mathcal{C}$), due to Cochran-Orr-Teichner, has been instrumental in the study of knot concordance in recent years. Part of its significance is due to the fact that certain geometric characterizations of a knot imply membership in various levels of the filtration. We show the counterpart of this fact for two new filtrations of $\mathcal{C}$ due to Cochran-Harvey-Horn, the positive and negative filtrations, denoted by $\{\mathcal{P}_n\}_{n=0}^\infty$ and $\{\mathcal{N}_n\}_{n=0}^\infty$ …
We use twisted Alexander polynomials to show that certain algebraically slice 2-bridge knots are not topologically slice, even though all prime power Casson-Gordon signatures vanish. We also provide some computations indicating the efficacy of Casson-Gordon signatures in obstructing the smooth sliceness of 2-bridge knots.
We give a useful classification of the metabelian unitary representations of pi_1(M_K), where M_K is the result of zero-surgery along a knot K in S^3. We show that certain eta invariants associated to metabelian representations pi_1(M_K) --> U(k) vanish for slice knots and that even more eta invariants vanish for ribbon knots and doubly slice knots. We show that our vanishing results contain the Casson-Gordon sliceness obstruction. In many cases eta invariants can be easily computed for satellite knots. We use this to study the relation between the eta invariant sliceness obstruction, the eta-…
We study chirally cosmetic surgeries, that is, a pair of Dehn surgeries on a knot producing homeomorphic 3-manifolds with opposite orientations. Several constraints on knots and surgery slopes to admit such surgeries are given. Our main ingredients are the original and the $SL(2,\mathbb{C})$ version of Casson invariants. As applications, we give a complete classification of chirally cosmetic surgeries on alternating knots of genus one.
Purely cosmetic surgeries and Casson--Walker--Lescop invariants
该研究证明了在有理同调球面中，无向同调结至多有两个整数纯 cosmetic 手术，并探讨了相关 3-流形中结的同胚性质。
We define an integer valued invariant for two-component links in S^3 by counting projective SU(2) representations of the link group having non-trivial second Stiefel-Whitney class. We show that our invariant is, up to sign, the linking number of the link. Our construction generalizes that of X.-S. Lin who defined a similar invariant for knots in S^3; his invariant equals half the knot signature.
We prove the additivity of the Casson-Seiberg-Witten invariant of integral homology $S^1 \times S^3$ under fiber sum along embedded curves and embedded tori, which is the $4$-dimensional analogue of the additivity of the Casson invariant under connected-sum and splicing along knots.
We derive formulae which lend themselves to TQFT interpretations of the Milnor torsion, the Lescop invariant, the Casson invariant, and the Casson-Morita cocyle of a 3-manifold, and, furthermore, relate them to the Reshetikhin-Turaev theory.
We compute the $SU(2)$ Casson-Lin invariant for the Hopf link and determine the sign in the formula of Harper and Saveliev relating this invariant to the linking number.
Given a rational homology sphere which bounds rational homology balls, we investigate the complexity of these balls as measured by the number of 1-handles in a handle decomposition. We use Casson-Gordon invariants to obtain lower bounds which also lead to lower bounds on the fusion number of ribbon knots. We use Levine-Tristram signatures to compute these bounds and produce explicit examples.
An $SL_2(\mathbb{R}) $-Casson invariant and Reidemeister torsions
该研究定义了闭3-流形的$SL_2(\mathbb{R})$-Casson不变量，给出其基于Reidemeister挠率的计算方法，还探讨了为该不变量赋予分次的途径。
Let $p$ be an odd prime and $G$ the finite cyclic group of order $p$. We use the Casson-Walker-Lescop invariant to find a necessary condition for a three-manifold to have an action of $G$ with a circle as the set of fixed points.
Twisted Blanchfield pairings and twisted signatures II: Relation to Casson-Gordon invariants
本文提出扭曲签名函数及其与Casson-Gordon不变量的关系，通过卫星公式扩展了结同调的研究。
In this paper we shall compute the Mrowka-Ruberman-Saveliev invariant introduced in [17] for the case when the manifold admits a free circle action.
We use intersection theory techniques to define an invariant of closed 3-manifolds counting the characters of irreducible representations of the fundamental group in PSL(2,C). We note several properties of the invariant and compute the invariant for certain Seifert fibered spaces and for some Dehn surgeries on twist knots. We discuss the relationship between this invariant and the SL(2,C)-invariant.
Gauss diagram formulas are extensively used to study Vassiliev link invariants. Now we apply this approach to invariants of 3-manifolds, considering manifolds given by surgery on framed links in the 3-sphere. We study the lowest degree case - the celebrated Casson-Walker invariant of rational homology spheres. This paper is dedicated to a detailed treatment of 2-component links; a general case will be considered in a forthcoming paper. We present simple Gauss diagram formulas for the Casson-Walker invariant. This enables us to understand/separate its dependence on the unframed link and on the …
We prove that the ($τ$-weighted, sheaf-theoretic) SL(2,C) Casson-Lin invariant introduced by Manolescu and the first author in [CM19] is generically independent of the parameter $τ$ and additive under connected sums of knots in integral homology 3-spheres. This addresses two questions asked in [CM19]. Our arguments involve a mix of topology, microlocal analysis and algebraic geometry, and rely crucially on the fact that the SL(2,C) Casson-Lin invariant admits an alternative interpretation via the theory of Behrend functions.
We provide a formula for the SU(3) Casson invariant for 3-manifolds given as the connected sum of two integral homology 3-spheres.
For any rational homology 3-sphere and one of its spin^{c}-structures, Ozsvath and Szabo defined a topological invariant, called d-invariant. Given a knot in the 3-sphere, the d-invariants associated with the prime-power-fold branched covers of the knot, obstruct the smooth sliceness of the knot. These invariants bear some structural resemblances to Casson-Gordon invariants, which obstruct the topological sliceness of a knot. Se-Goo Kim found a polynomial splitting property for Casson-Gordon invariants. In this paper, we show a similar result for Ozsvath and Szabo's d-invariants. We give an ap…
We establish the exact triangle in Seiberg-Witten-Floer theory relating the monopoloe homologies of any two closed 3-manifolds which are obtained from each other by $\pm 1$-surgery. We also show that the sum of the modified version of the Seiberg-Witten invariants for any closed rational homology 3-sphere $Y$ over all $Spin^c$ structures equals to $\frac 12 |H_1(Y, \Z)| λ(Y)$ where $λ(Y)$ is the Casson-Walker invariant.
On the Casson Invariant Conjecture of Neumann--Wahl
本文证明了Neumann-Wahl关于splice类型表面奇点的Casson不变量猜想，指出链环的Casson不变量等于Milnor纤维签名的八分之一。
Suppose Y is an integer homology 3-sphere, Taubes proved that the number of irreducible critical orbits of the perturbed Chern-Simons functional on Y, counted with signs, is equal to the algebraic intersection number of two character varieties associated with Heegaard splittings when the structure group is SU(2). Taubes' result established a relationship between gauge theory and the Casson invariant. This article proves an analogous identification result for SU(n) generalized Casson invariants. As a special case, we show that the SU(3) Casson invariant of Boden-Herald can be equivalently calcu…
Formulas previously presented for the Casson-Walker invariant are generalized to Lescop's extension. These formulas in terms of linking numbers and surgery coefficients compute the change in Lescop's invariant under crossing changes in a framed link presenting a 3-manifold. This leads us to revisit an old formula for a coefficient of the Conway polynomial. Finally we compute Lescop's invariant of several lens spaces and deduce some values for Dedekind sums.
It is known that every oriented integral homology 3-sphere can be obtained from S^3 by a finite sequence of Borromean surgeries. We give an explicit formula for the variation of the Casson invariant under such a surgery move. The formula involves simple classical invariants, namely the framing, linking number and Milnor's triple linking number. A more general statement, for n independent Borromean surgeries, is also provided, which involves an additional cubic expression in some linking numbers of the surgery link.
We prove a surgery formula of the Casson-Seiberg-Witten invariant of integral homology $S^1 \times S^3$ along an embedded torus, which could either be regarded as an extension of the product formula for Seiberg-Witten invariants or a manifestation of the surgery exact triangle in $4$-dimensional Seiberg-Witten theory of homology $S^1 \times S^3$. As an application, we compute this invariant for mapping tori of $3$-manifolds under diffeomorphisms of finite order and fixed-point set being a simple closed curve. This computation generalizes the result of Lin-Ruberman-Saveliev.
A perturbative SU(3) Casson invariant $Λ_{SU(3)}(X)$ for integral homology 3-spheres is defined. Besides being fully perturbative, it has nice properties: (1) $4 . Λ_{SU(3)}(X)$ is an integer. (2) It is preseved under orientation change. (3) A connected sum formula holds. Explicit calculations of the invariant for $1/k$ surgery on $(2,q)$ torus knots are made and compared with a different generalization of Casson's invariant to SU(3) by Boden and Herald. For those cases computed, the invariant defined here is a quadratic polynomial in k for k>0 and a quadratic polynomial for k<0.
Casson-type invariants emerging from Donaldson theory over certain negative definite 4-manifolds were recently suggested by Andrei Teleman. These are defined by a count of a zero-dimensional moduli space of flat instantons. Motivated by the cobordism program of proving Witten's conjecture, we use a moduli space of PU(2) Seiberg-Witten monopoles to exhibit an oriented one-dimensional cobordism of the instanton moduli space to the empty space. The Casson-type invariant must therefore vanish.
We introduce a multivariable Casson-Lin type invariant for links in $S^3$. This invariant is defined as a signed count of irreducible $\operatorname{SU}(2)$ representations of the link group with fixed meridional traces. For 2-component links with linking number one, the invariant is shown to be a sum of multivariable signatures. We also obtain some results concerning deformations of $\operatorname{SU}(2)$ representations of link groups.
Updated rerefences and introduction. Given a knot in an integer homology sphere, one can construct a family of closed 3-manifolds (parametrized by the positive integers), namely the cyclic branched coverings of the knot. In this paper we give a formula for the the Casson-Walker invariants of these 3-manifolds in terms of residues of a rational function (which measures the 2-loop part of the Kontsevich integral of a knot) and the signature function of the knot. Our main result actually computes the LMO invariant of cyclic branched covers in terms of a rational invariant of the knot and its sign…
We introduce the $SU(N)$ Casson-Lin invariants for links $L$ in $S^3$ with more than one component. Writing $L = \ell_1 \cup \cdots \cup \ell_n$, we require as input an $n$-tuple $(a_1,\ldots, a_n) \in {\mathbb Z}^n$ of labels, where $a_j$ is associated with $\ell_j$. The $SU(N)$ Casson-Lin invariant, denoted $h_{N,a}(L)$, gives an algebraic count of certain projective $SU(N)$ representations of the link group $π_1(S^3 \smallsetminus L)$, and the family $h_{N,a}$ of link invariants gives a natural extension of the $SU(2)$ Casson-Lin invariant, which was defined for knots by X.-S. Lin and for 2…
We define an integer valued invariant of homology spheres using the methods of SU(3) gauge theory and study its behavior under orientation reversal and connected sum.
In knot concordance three genera arise naturally, g(K), g_4(K), and g_c(K): these are the classical genus, the 4-ball genus, and the concordance genus, defined to be the minimum genus among all knots concordant to K. Clearly 0 <= g_4(K) <= g_c(K) <= g(K). Casson and Nakanishi gave examples to show that g_4(K) need not equal g_c(K). We begin by reviewing and extending their results. For knots representing elements in A, the concordance group of algebraically slice knots, the relationships between these genera are less clear. Casson and Gordon's result that A is nontrivial implies that g_4(K) ca…
A unified Casson-Lin invariant for the real forms of SL(2)
本文提出了一种统一的Casson-Lin不变量，用于研究SL(2,R)表示与SU(2)表示的关系，并证明了其通过拓扑假设可推导出多种表示的存在性。
Universal invariants, the Conway polynomial and the Casson-Walker-Lescop invariant
本文通过图示和组合方法给出了闭3流形卡森-沃克-莱斯科普不变量的手术公式，揭示了康威多项式系数与 Kontsevich 积分系数的关系。
In this paper, we prove a conjecture of Friedl and Powell that their Casson-Gordon type invariant of 2-component link with linking number one is actually an obstruction to being height 3.5 Whitney tower/grope concordant to the Hopf Link. The proof employs the notion of solvable cobordism of 3-manifolds with boundary, which was introduced by Cha. We also prove that the Blanchfield form and the Alexander polynomial of links in $S^3$ give obstructions to height 3 Whitney tower/grope concordance. This generalizes the results of Hillman and Kawauchi.
This is a survey of our recent work with Tom Mrowka on Seiberg-Witten gauge theory and index theory for manifolds with periodic ends. We explain how this work leads to a new invariant, which is related to the classical Rohlin invariant of homology 3-spheres and to the Furuta-Ohta invariant originating in Yang-Mills gauge theory. We give some new calculations of our invariant for 4-dimensional mapping tori.
We use Heegaard decompositions and the theta divisor on a Riemannian surface to define a three-manifold invariant for rational homology three-spheres. This invariant is defined on the set of $Spin^c$ structures $$ {\hat θ}\colon Spin^c(Y)\longrightarrow Q. $$ In the first part of the paper, we give the definition of the invariant (which builds on the theory developed in the article, "The theta divisor and three-manifold invariants"). In the second part, we establish a relationship between this invariant and the Casson-Walker invariant.
This article surveys our ongoing project about the relationship between invariants extending the classical Rohlin invariant of homology spheres and those coming from 4-dimensional (Yang-Mills) gauge theory. The main conjecture towards which this project is directed is that the Rohlin invariant and the gauge theoretic invariant coincide for a homology $S^1\times S^3$. We discuss the implications of this conjecture for some classical problems in low-dimensional topology, and progress we have made towards proving the conjecture.
Let Z^{LMO} be the 3-manifold invariant of [LMO]. It is shown that Z^{LMO}(M)=1, if the first Betti number of M, b_{1}(M), is greater than 3. If b_{1}(M)=3, then Z^{LMO}(M) is completely determined by the cohomology ring of M. A relation of Z^{LMO} with the Rozansky-Witten invariants Z_{X}^{RW}[M] is established at a physical level of rigour. We show that Z_{X}^{RW}[M] satisfies appropriate connected sum properties suggesting that the generalized Casson invariant ought to be computable from the LMO invariant.
We develop techniques for computing the integer valued SU(3) Casson invariant. Our method involves resolving the singularities in the flat moduli space using a twisting perturbation and analyzing its effect on the topology of the perturbed flat moduli space. These techniques, together with Bott-Morse theory and the splitting principle for spectral flow, are applied to calculate the invariant for all Brieskorn homology spheres.
Klein-four connections and the Casson invariant for non-trivial admissible $U(2)$ bundles
本文研究非平凡容许 $U(2)$ 丛的 Casson 不变量，通过模2上同调环及 Klein 四群完整群平坦联络的计数公式，刻画其偶性来源。
We derive a gauge theoretic invariant of integral homology 3-spheres which counts gauge orbits of irreducible, perturbed flat SU(3) connections with sign given by spectral flow. To compensate for the dependence of this sum on perturbations, the invariant includes contributions from the reducible, perturbed flat orbits. Our formula for the correction term generalizes that given by Walker in his extension of Casson's SU(2) invariant to rational homology 3-spheres.
We bound the value of the Casson invariant of any integral homology 3-sphere $M$ by a constant times the distance-squared to the identity, measured in any word metric on the Torelli group $\T$, of the element of $\T$ associated to any Heegaard splitting of $M$. We construct examples which show this bound is asymptotically sharp.
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