Narrow Results

As an extension of Reshetikhin and Turaev’s invariant, Costantino, Geer and Patureau-Mirand constructed $3$-manifold invariants in the setting of relative $G$-modular categories, which include both semi-simple and non-semi-simple ribbon tensor categories as examples. In this paper, we follow their method to construct a $3$-manifold invariant from Viro’s $𝔤𝔩(1|1)$-Alexander polynomial. We take lens spaces $L(7,1)$ and $L(7,2)$ as examples to show that this invariant can distinguish homotopy equivalent manifolds.

A central problem in low-dimensional topology asks which homology three-spheres bound contractible four-manifolds or homology four-balls. In this paper, we address this question for plumbed three-manifolds and we present two new infinite families. We consider most of the classical examples from around the 1980s by reproving that they all bound Mazur manifolds. We also show that several well-known families bound possibly different types of four-manifolds, called Poénaru homology four-balls. To unify classical and new results in a fairly simple way, we modify Mazur’s argument and work with Poénaru manifolds.

We complete the project begun by Callahan, Dean and Weeks to identify all knots whose complements are in the SnapPea census of hyperbolic manifolds with seven or fewer tetrahedra. Many of these "simple" hyperbolic knots have high crossing number. We also compute their Jones polynomials.

Akbulut generalized Fintushel–Stern knot surgery in a certain way. He showed that an operation C(K#(-K)) → {diffeo type of X} is not injective, where C(K#(-K)) is self-concordance set of K#(-K) and K is trefoil knot. In the present paper we prove that the same phenomenon occurs for any knot K and for some other knots L instead of K#(-K).

It is shown that any closed three-manifold M obtained by integral surgery on a knot in the three-sphere can always be constructed from integral surgeries on a 3-component link with each component being an unknot in the three-sphere. It is also interesting to notice that infinitely many different integral surgeries on the same link could give the same three-manifold M.

We show that for any n ≥ 4 there exists an equivalence functor from the category of n-fold connected simple coverings of B³ × [0, 1] branched over ribbon surface tangles up to certain local ribbon moves, and the cobordism category of orientable relative 4-dimensional 2-handlebody cobordisms up to 2-deformations. As a consequence, we obtain an equivalence theorem for simple coverings of S³ branched over links, which provides a complete solution to the long-standing Fox–Montesinos covering moves problem. This last result generalizes to coverings of any degree results by the second author and Apostolakis, concerning respectively the case of degree 3 and 4. We also provide an extension of the equivalence theorem to possibly non-simple coverings of S³ branched over embedded graphs. Then, we factor the functor above as , where is an equivalence functor to a universal braided category freely generated by a Hopf algebra object H. In this way, we get a complete algebraic description of the category . From this we derive an analogous description of the category of 2-framed relative 3-dimensional cobordisms, which resolves a problem posed by Kerler.

We introduce new topological quantum invariants, surface link quantum invariants, of compact oriented 3-manifolds with boundary where the boundary is a disjoint union of two identical surfaces. The invariants are constructed by using virtual knots and links.

These invariants are new, nontrivial, and calculable examples of quantum invariants of 3-manifolds with non-vacuous boundary.

Our new invariants, surface link quantum invariants, give new invariants of classical knots and links, i.e., knots and links in the 3-sphere. That is, virtual knot theory induces a new, classical knot invariant.

Harer, Kas and Kirby conjectured that every handle decomposition of the Dolgachev surface E(1)_2,3 requires both 1- and 3-handles. In this article, we construct a smooth 4-manifold which has the same Seiberg-Witten invariant as E(1)_2,3 and has neither 1- nor 3-handles in a handle decomposition.