Upsilon invariants of L-space cable knots

We compute the Upsilon invariant of L-space cable knots $K_{p,q}$ in terms of $p,\Upsilon_K$ and $\Upsilon_{T_{p,q}}$. The integral value of the Upsilon invariant gives a ${\mathbb Q}$-valued knot concordance invariant. We also compute the integral values of the Upsilon of L-space cable knots.Comment: 21 pages, some figure

Heegaard Floer homology of Matsumoto's manifolds

We consider a homology sphere $M_n(K_1,K_2)$ presented by two knots $K_1,K_2$ with linking number 1 and framing $(0,n)$. We call the manifold {\it Matsumoto's manifold}. We show that there exists no contractible bound of $M_n(T_{2,3},K_2)$ if $n<2\tau(K_2)$ holds. We also give a formula of Ozsv\'ath-Szab\'o's $\tau$-invariant as the total sum of the Euler numbers of the reduced filtration. We compute the $\delta$-invariants of the twisted Whitehead doubles of torus knots and correction terms of the branched covers of the Whitehead doubles. By using Owens and Strle's obstruction we show that the $12$-twisted Whitehead double of the $(2,7)$-torus knot and the $20$-twisted Whitehead double of the $(3,7)$-torus knot are not slice but the double branched covers bound rational homology 4-balls. These are the first examples having a gap between sliceness and rational 4-ball bound-ness of the double branched cover.Comment: 22 pages, 15 figures, 1 tabl

A complete list of lens spaces constructed by Dehn surgery I

Berge in [1] defined doubly primitive knots, which yield lens spaces by Dehn surgery. At the same paper he listed the knots into several types. In this paper we will prove the list is complete when $\tau>1$. The invariant $\tau$ is a quantity with regard to lens space surgery, which is defined in this paper. Furthermore at the same time we will also prove that Table~6 in [8] is complete as Poincar\'e homology sphere surgery when $\tau>1$.Comment: 46 pages, 41 figures and 4 table

Boundary-sum irreducible finite order corks

We prove for any positive integer $n$ there exist boundary-sum irreducible ${\mathbb Z}_n$-corks with Stein structure. Here `boundary-sum irreducible' means the manifold is indecomposable with respect to boundary-sum. We also verify that some of the finite order corks admit hyperbolic boundary by HIKMOT.Comment: 11 pages, 7 figure

On the non-existence of L-space surgery structure

We exhibit homology spheres which never yield lens spaces by any integral Dehn surgery by using Ozsvath Szabo's contact invariant.Comment: 5 page

On the Alexander polynomial of lens space knot

Ozsv\'ath-Szab\'o proved the property that any coefficient of Alexander polynomial of lens space knot is either $\pm1$ or $0$ and the non-zero coefficients are alternating. Combining the formulas of the Alexander polynomial of lens space knots due to Kadokami-Yamada and Ichihara-Saito-Teragaito, we refine Ozsv\'ath-Szab\'o's property as the existence of simple curves included in a region in ${\Bbb R}^2$. The existence of curves, that has no end-points connected, is just 1-component in a region, can search distribution of non-zero coefficients of the Alexander polynomial of the lens space knot. This curve is much useful to obtain constraints of Alexander polynomials of lens space knots. For example, we can investigate the location of the second, third and fourth non-zero coefficients. The curve extracts new invariant $\alpha$-index. The invariant is an important factor to determine Alexander polynomial of lens space knot. We classify lens space surgeries that the Alexander polynomial is the same as a $(2,r)$-torus knot and lens space surgeries with small genus and so on. As well as lens space knots in $S^3$, we also deal with lens space knots in homology spheres, which the surgery duals are simple (1,1)-knots.Comment: 38 pages, 22 figures, 3 table

Lens spaces given from L-space homology 3-spheres

We consider the problem when lens spaces are given from homology spheres, and demonstrate that many lens spaces are obtained from L-space homology sphere which the Ozsv\'ath Szab\'o's correction term $d(Y)$ is equal to 2. We show an inequality of slope and genus when $Y$ is L-space and $Y_p(K)$ is lens space.Comment: 11 pages, 2 tables, 2 figure

Variations of 4-dimensional twists obtained by an infinite order plug

In the previous paper the author defined an infinite order plug $(P,\varphi)$ which gives rise to infinite Fintushel-Stern's knot-surgeries. Here, we give two 4-dimensional infinitely many exotic families $Y_n$, $Z_n$ of exotic enlargements of the plug. The families $Y_n$, $Z_n$ have $b_2=3$, $4$ and the boundaries are 3-manifolds with $b_1=1$, $0$ respectively. We give a plug (or g-cork) twist $(P,\varphi_{p,q})$ producing the 2-bridge knot or link surgery by combining the plug $(P,\varphi)$. As a further example, we describe a 4-dimensional twist $(M,\mu)$ between knot-surgeries for two mutant knots. The twisted double concerning $(M,\mu)$ gives a candidate of exotic $\#^2S^2\times S^2$.Comment: 31 pages, 33 figure

The E8E_8-boundings of homology spheres and negative sphere classes in E(1)E(1)

We define invariants $\frak{ds}$ and $\overline{\frak{ds}}$, which are the maximal and minimal second Betti number divided by $8$ among definite spin boundings of a homology sphere. The similar invariants $g_8$ and $\overline{g_8}$ are defined by the maximal (or minimal) product sum of $E_8$-form of bounding 4-manifolds. We compute these invariants for some homology spheres. We construct $E_8$-boundings for some of Brieskorn 3-spheres $\Sigma(2,3,12n+5)$ by handle decomposition. As a by-product of the construction, some negative classes which consist of addition of several fiber classes plus one sectional class in $E(1)$ are represented by spheres.Comment: 24 pages, 16 figure

Homology spheres yielding lens spaces

It is known by the author that there exist 20 families of Dehn surgeries in the Poincar\'e homology sphere yielding lens spaces. In this paper, we give the concrete knot diagrams of the families and extend them to families of lens space surgeries in Brieskorn homology spheres. We illustrate families of lens space surgeries in $\Sigma(2,3,6n\pm1)$ and $\Sigma(2,2s+1,2(2s+1)n\pm1)$ and so on. As other examples, we give lens space surgeries in graph homology spheres, which are obtained by splicing two Brieskorn homology spheres.Comment: 40 pages and 45 figures, to appear in Proceedings of the Gokova Geometry-Topology Conference 201