Jae Choon Cha

An Identity-Based Signature from Gap Diffie-Hellman Groups

In this paper we propose an identity(ID)-based signature scheme using gap Diffie-Hellman (GDH) groups. Our scheme is proved secure against existential forgery on adaptively chosen message and ID attack under the random oracle model. Using GDH groups obtained from bilinear pairings, as a special case of our scheme, we obtain an ID-based signature scheme that shares the same system parameters with the ID-based encryption scheme (BF-IBE) by Boneh and Franklin [BF01], and is as efficient as the BF-IBE. Combining our signature scheme with the BF-IBE yields a complete solution of an ID-based public key system. It can be an alternative for certificate-based public key infrastructures, especially when efficient key management and moderate security are required.

Fibred knots and twisted Alexander invariants

We study the twisted Alexander invariants of fibred knots. We establish necessary conditions on the twisted Alexander invariants for a knot to be fibred, and develop a practical method to compute the twisted Alexander invariants from the homotopy type of a monodromy. It is illustrated that the twisted Alexander invariants carry more information on fibredness than the classical Alexander invariants, even for knots with trivial Alexander polynomials.

Algebraic and Heegaard–Floer invariants of knots with slice Bing doubles

If the Bing double of a knot K is slice, then K is algebraically slice. In addition the Heegaard-Floer concordance invariants �, developed by Ozsvath-Szabo, and �, developed by Manolescu and Owens, vanish on K. For a knot K ⊂ S 3 , the Bing double, denoted B(K), is the two component link illustrated schematically in Figure 1. Within the box the two strands run parallel along a diagram for K, and for this to be well-defined, independent of the choice of diagram of K, the strands are twisted so that their algebraic crossing number within the box is zero.

Link concordance, homology cobordism, and Hirzebruch-type defects from iterated p-covers

We obtain new invariants of topological link concordance and homology cobordism of 3-manifolds from Hirzebruch-type intersection form defects of towers of iterated p-covers. Our invariants can extract geometric information from an arbitrary depth of the derived series of the fundamental group, and can detect torsion which is invisible via signature invariants. Applications illustrating these features include the following: (1) There are infinitely many homology equivalent rational 3-spheres which are indistinguishable via multisignatures, eta-invariants, and L2-signatures but have distinct homology cobordism types. (2) There is an infinite family of 2-torsion (amphichiral) knots, including the figure eight knot, with non-slice iterated Bing doubles; as a special case, we give the first proof of the conjecture that the Bing double of the figure eight knot is not slice. (3) There exist infinitely many torsion elements at any depth of the Cochran-Orr-Teichner filtration of link concordance.

The cobordism group of homology cylinders

Garoufalidis and Levine introduced the homology cobordism group of homology cylinders over a surface. This group can be regarded as an enlargement of the mapping class group. Using torsion invariants, we show that the abelianization of this group is infinitely generated provided that the first Betti number of the surface is positive. In particular, this shows that the group is not perfect. This answers questions of Garoufalidis and Levine, and Goda and Sakasai. Furthermore, we show that the abelianization of the group has infinite rank for the case that the surface has more than one boundary component. These results also hold for the homology cylinder analogue of the Torelli group.

Knot signature functions are independent

A Seifert matrix is a square integral matrix V satisfying det(V-V T )=±1. To such a matrix and unit complex number ω there corresponds a signature, σ ω (V) = sign((1 - ω)V + (1 - ω)V T ). Let S denote the set of unit complex numbers with positive imaginary part. We show that {σ ω } ω ∈ S is linearly independent, viewed as a set of functions on the set of all Seifert matrices. If V is metabolic, then σ ω (V) = 0 unless ω is a root of the Alexander polynomial, Δ V (t) = det(V - tV T ). Let A denote the set of all unit roots of all Alexander polynomials with positive imaginary part. We show that {σ ω } ω ∈ A is linearly independent when viewed as a set of functions on the set of all metabolic Seifert matrices. To each knot K C S 3 one can associate a Seifert matrix V K , and σ ω (V K ) induces a knot invariant. Topological applications of our results include a proof that the set of functions {σ ω } ω ∈ S is linearly independent on the set of all knots and that the set of two-sided averaged signature functions, {σ* ω } ω ∈ S , forms a linearly independent set of homomorphisms on the knot concordance group. Also, if v ∈ S is the root of some Alexander polynomial, then there is a slice knot K whose signature function σ ω (K) is nontrivial only at ω = v and ω = v. We demonstrate that the results extend to the higher-dimensional setting.

Signatures of links in rational homology spheres

Abstract A theory of signatures for odd-dimensional links in rational homology spheres is studied via their generalized Seifert surfaces. The jump functions of signatures are shown invariant under appropriately generalized concordance and a special care is given to accommodate one-dimensional links with mutual linking. Furthermore our concordance theory of links in rational homology spheres remains highly nontrivial after factoring out the contribution from links in integral homology spheres.

Symmetric Whitney tower cobordism for bordered 3-manifolds and links

We introduce a notion of symmetric Whitney tower cobordism between bordered 3-manifolds, aiming at the study of homology cobordism and link concordance. It is motivated by the symmetric Whitney tower approach to slicing knots and links initiated by Cochran, Orr, and Teichner. We give amenable Cheeger-Gromov rho-invariant obstructions to bordered 3-manifolds being Whitney tower cobordant. Our obstruction is related to and generalizes several prior known results, and also gives new interesting cases. As an application, our method applied to link exteriors reveals new structures on (Whitney tower and grope) concordance between links with nonzero linking number, including the Hopf link.

Signature invariants of links from irregular covers and non-abelian covers

Signature invariants of odd dimensional links from irregular covers and nonabelian covers of complements are obtained by using the technique of Casson and Gordon. We show that the invariants vanish for slice links and can be considered as invariants under Fm-link concordance. We illustrate examples of links that are not slice but behave as slice links for any invariants from abelian covers.

On equivariant slice knots

We suggest a method to detect that two periodic knots are not equivariantly concordant, using surgery on factor links. We construct examples which satisfy all known necessary conditions for equivariant slice knotsNaiks and Choi-Ko-Songs improvements of classical results on Seifert forms and Casson-Gordon invariants of slice knots but are not equivariantly slice.