Bridge position and the representativity of spatial graphs

In this paper, we consider closed surfaces which contain spatial graphs. In the case that a closed surface is a 2-sphere, we show that the 2-sphere can be isotoped so that it intersects a bridge sphere for the spatial graph in a single loop. In the case that a closed surface is not a 2-sphere, we define an invariant of a spatial graph by counting the number of intersection of a compressing disk for the closed surface and the spatial graph. By using this invariant, we give a lower bound for the bridge number of a spatial graph.     

On the localization formula in equivariant cohomology

We give a generalization of the Atiyah-Bott-Berline-Vergne localization theorem for the equivariant cohomology of a torus action. We replace the manifold having a torus action by an equivariant map of manifolds having a compact connected Lie group action. This provides a systematic method for calculating the Gysin homomorphism in ordinary cohomology of an equivariant map. As an example, we recover a formula of Akyildiz-Carrell for the Gysin homomorphism of flag manifolds.     

Special manifolds and shape fibrator properties

Our main interest in this paper is further investigation of the concept of (PL) fibrators (introduced by Daverman [R.J. Daverman, PL maps with manifold fibers, J. London Math. Soc. (2) 45 (1992) 180-192]), in a slightly different PL setting. Namely, we are interested in manifolds that can detect approximate fibrations in the new setting. The main results state that every orientable, special (a new class of manifolds that we introduce) PL n-manifold with non-trivial first homology group is a fibrator in the new category, if it is a codimension-2 fibrator (Theorem 8.2) or has a non-cyclic fundamental group (Theorem 8.4). We show that all closed, orientable surface S with χ(S)<0 are fibrators in the new category.     

On minimal elements for a partial order of prime knots

In this paper, applying Chebyshev polynomials we give a basic proof of the irreducibility over the complex number field of the defining polynomial of SL₂(C)-character variety of twist knots in infinitely many cases. The irreducibility, combined with a result in the paper of M. Boileau, S. Boyer, A.W. Reid and S. Wang in 2010, shows the minimality of infinitely many twist knots for a partial order on the set of prime knots defined by using surjective group homomorphisms between knot groups. In Appendix B, we also give a straightforward proof of the result of Boileau et al.     

The complete splitting number of a lassoed link

In this paper we define a lassoing on a link, a local addition of a trivial knot to a link. Let K be an s-component link with the Conway polynomial non-zero. Let L be a link which is obtained from K by r-iterated lassoings. The complete splitting number split(L) is greater than or equal to r+s−1, and less than or equal to r+split(K). In particular, we obtain from a knot by r-iterated component-lassoings an algebraically completely splittable link L with split(L)=r. Moreover, we construct a link L whose unlinking number is greater than split(L).     

Polynomial invariants and quandles of twisted links

Bourgoin defined the notion of a twisted link which corresponds to a stable equivalence class of links in oriented thickenings. It is a generalization of a virtual link. Some invariants of virtual links are extended for twisted links including the knot group and the Jones polynomial. In this paper, we generalize a multivariable polynomial invariant of a virtual link to a twisted link. We also introduce a quandle of a twisted link.     

Closed incompressible surfaces of genus two in 3-bridge knot complements

In this paper, we characterize closed incompressible surfaces of genus two in the complements of 3-bridge knots and links. This characterization includes that of essential 2-string tangle decompositions for 3-bridge knots and links.     

On the quantum filtration of the Khovanov-Rozansky cohomology

We prove the quantum filtration on the Khovanov-Rozansky link cohomology Hp with a general degree (n+1) monic potential polynomial p(x) is invariant under Reidemeister moves, and construct a spectral sequence converging to Hp that is invariant under Reidemeister moves, whose E₁ term is isomorphic to the Khovanov-Rozansky sl(n)-cohomology Hn. Then we define a generalization of the Rasmussen invariant, and study some of its properties. We also discuss relations between upper bounds of the self-linking number of transversal links in standard contact S³.     

Some relations on Miyazawa's virtual knot invariant

Y. Miyazawa defined a polynomial invariant for a virtual link by using magnetic graph diagrams, which is related with the Jones-Kauffman polynomial. In this paper we show some relations of this polynomial for a virtual skein triple.     

Twisted Alexander polynomials of twist knots for nonabelian representations

In this paper, we describe the twisted Alexander polynomial of twist knots for nonabelian SL(2,C)-representations and investigate in detail the coefficient of the highest degree term as a function on the representation space of the knot group. In particular, we introduce the notion of monic representation and discuss its relation to the fiberedness of knots.     

Rational structure on algebraic tangles and closed incompressible surfaces in the complements of algebraically alternating knots and links

Let F be an incompressible, meridionally incompressible and not boundary-parallel surface with boundary in the complement of an algebraic tangle (B,T). Then F separates the strings of T in B and the boundary slope of F is uniquely determined by (B,T) and hence we can define the slope of the algebraic tangle. In addition to the Conway's tangle sum, we define a natural product of two tangles. The slopes and binary operation on algebraic tangles lead to an algebraic structure which is isomorphic to the rational numbers.We introduce a new knot and link class, algebraically alternating knots and links, roughly speaking which are constructed from alternating knots and links by replacing some crossings with algebraic tangles. We give a necessary and sufficient condition for a closed surface to be incompressible and meridionally incompressible in the complement of an algebraically alternating knot or link K. In particular we show that if K is a knot, then the complement of K does not contain such a surface.     

Hyperbolic isometries versus symmetries of links

We prove that every finite group is the orientation-preserving isometry group of the complement of a hyperbolic link in the 3-sphere.     

Crossing distances of surface-knots

A surface-knot is an embedded closed connected oriented surface in 4-space. A surface diagram is a projection of a surface-knot into 3-space with crossing information. In this paper we define a distance from a special surface diagram to a trivial diagram as the minimal number of special double cycles, where we can change the crossing information to obtain the trivial diagram. We estimate the distance using the number of 1-handles needed to obtain a trivial diagram.     

A refinement of the Bernštein-Kušnirenko estimate

A theorem of Kušnirenko and Bernštein (also known as the BKK theorem) shows that the number of isolated solutions in a torus to a system of polynomial equations is bounded above by the mixed volume of the Newton polytopes of the given polynomials, and this upper bound is generically exact. We improve on this result by introducing refined combinatorial invariants of polynomials and a generalization of the mixed volume of convex bodies: the mixed integral of concave functions. The proof is based on new techniques and results from relative toric geometry.     

An alternative proof of Lickorish–Wallace theorem

We introduce the concept of s-distance of an unstabilized Heegaard splitting. We prove if a 3-manifold admits an unstabilized genus g Heegaard splitting with s-distance m  , then surgery on some (m−1)$(m-1)$ components link may produce a 3-manifold which admits a stabilized genus g Heegaard splitting. We also give an alternative proof of the fundamental theorem of surgery theory, which states that every closed orientable 3-manifold is obtained by surgery on some link in 3-sphere.     

A personal account of the discovery of hyperbolic structures on some knot complements

I give my view of the early history of the discovery of hyperbolic structures on knot complements from my early work on representations of knot groups into matrix groups to my meeting with William Thurston in 1976.     

Noncyclic Covers of Knot Complements

Hempel has shown that the fundamental groups of knot complements are residually finite. This implies that every nontrivial knot must have a finite-sheeted, noncyclic cover. We give an explicit bound, Φ (c), such that if K is a nontrivial knot in the three-sphere with a diagram with c crossings then the complement of K has a finite-sheeted, noncyclic cover with at most Φ (c) sheets.The author is supported by an NSF Postdoctoral Fellowship at Cornell University.     

On a surface singular braid monoid

We introduce a monoid corresponding to knotted surfaces in four space, from its hyperbolic splitting represented by marked diagram in braid like form. It has four types of generators: two standard braid generators and two of singular type. Then we state relations on words that follow from topological Yoshikawa moves. As a direct application we will reprove some known theorem about twist-spun knots. We wish then to investigate an index associated to the closure of surface singular braid. Using our relations we will prove that there are exactly six types of knotted surfaces with the index less or equal to two, and there are infinitely many types of surface-knots with index equal to three. Towards the end we will construct a family of classical diagrams such that to unlink them requires at least four Reidemeister III moves.