Crosscap numbers of 2-bridge knots

We present a practical algorithm to determine the minimal genus of non-orientable spanning surfaces for 2-bridge knots, called the crosscap numbers. We will exhibit a table of crosscap numbers of 2-bridge knots up to 12 crossings (all 362 of them).     

Annuli in generalized Heegaard splittings and degeneration of tunnel number

Abstract. We analyze how a family of essential annuli in a compact 3-manifold will induce, from a strongly irreducible generalized Heegaard splitting of the ambient manifold, generalized Heegaard splittings of the complementary components. There are specific applications to the subadditivity of tunnel number of knots, improving somewhat bounds of Kowng [Kw]. For example, in the absence of 2-bridge summands, the tunnel number of the sum of n knots is no less than the sum of the tunnel numbers. Received: 10 November 1999 / Published online: 28 June 2000     

Heegaard splittings of twisted torus knots

Little is known on the classification of Heegaard splittings for hyperbolic 3-manifolds. Although Kobayashi gave a complete classification of Heegaard splittings for the exteriors of 2-bridge knots, our knowledge of other classes is extremely limited. In particular, there are very few hyperbolic manifolds that are known to have a unique minimal genus splitting. Here we demonstrate that an infinite class of hyperbolic knot exteriors, namely exteriors of certain “twisted torus knots” originally studied by Morimoto, Sakuma and Yokota, have a unique minimal genus Heegaard splitting of genus two. We also conjecture that these manifolds possess irreducible yet weakly reducible splittings of genus three. There are no known examples of such Heegaard splittings.     

q-holonomic formulas for colored HOMFLY polynomials of 2-bridge links

We compute q-holonomic formulas for the HOMFLY polynomials of 2-bridge links colored with one-column (or one-row) Young diagrams.     

Finite type invariants of nanowords and nanophrases

Homotopy classes of nanowords and nanophrases are combinatorial generalizations of virtual knots and links. Goussarov, Polyak and Viro defined finite type invariants for virtual knots and links via semi-virtual crossings. We extend their definition to nanowords and nanophrases. We study finite type invariants of low degrees. In particular, we show that the linking matrix and T invariant defined by Fukunaga are finite type of degree 1 and degree 2 respectively. We also give a finite type invariant of degree 4 for open homotopy of Gauss words.     

A refinement of the Conway-Gordon theorems

In 1983, Conway-Gordon showed that for every spatial complete graph on 6 vertices, the sum of the linking numbers over all of the constituent 2-component links is congruent to 1 modulo 2, and for every spatial complete graph on 7 vertices, the sum of the Arf invariants over all of the Hamiltonian knots is also congruent to 1 modulo 2. In this article, we give integral lifts of the Conway-Gordon theorems above in terms of the square of the linking number and the second coefficient of the Conway polynomial. As applications, we give alternative topological proofs of theorems of Brown-Ramírez Alfonsín and Huh-Jeon for rectilinear spatial complete graphs which were proved by computational and combinatorial methods.     

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.     

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.     

On virtual pathwidth of virtual graphs of a virtual link

The geometric representation of a knot is not too dissimilar from a graph and this interaction has helped mathematicians to solve many problems. In this paper, we apply graph theory tools to study the classification of virtual knots and links. We define virtual planar graphs and compute virtual path width of an associated graph of a virtual link. We show that the virtual path width of an associated graph is equal to the virtual bridge number of a pseudo prime knot.     

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.     

Vassiliev invariants and the Poincaré conjecture

This article examines the relationship between 3-manifold topology and knot invariants of finite type. We prove that in every Whitehead manifold there exist knots that cannot be distinguished by Vassiliev invariants. If, on the other hand, Vassiliev invariants distinguish knots in each homotopy sphere, then the Poincaré conjecture is true (i.e. every homotopy 3-sphere is homeomorphic to the standard 3-sphere).     

On the Alexander polynomials of knots with Gordian distance one

We consider a condition on a pair of the Alexander polynomials of knots which are realizable by a pair of knots with Gordian distance one. We show that there are infinitely many mutually disjoint infinite subsets in the set of the Alexander polynomials of knots such that every pair of distinct elements in each subset is not realizable by any pair of knots with Gordian distance one. As one of the subsets, we have an infinite set containing the Alexander polynomials of the trefoil knot and the figure eight knot. We also show that every pair of distinct Alexander polynomials such that one is the Alexander polynomial of a slice knot is realizable by a pair of knots of Gordian distance one, so that every pair of distinct elements in the infinite subset consisting of the Alexander polynomials of slice knots is realizable by a pair of knots with Gordian distance one. These results solve problems given by Y. Nakanishi and by I. Jong.     

New obstructions to doubly slicing knots

A knot in the 3-sphere is called doubly slice if it is a slice of an unknotted 2-sphere in the 4-sphere. We give a bi-sequence of new obstructions for a knot being doubly slice. We construct it following the idea of Cochran-Orr-Teichner's filtration of the classical knot concordance group. This yields a bi-filtration of the monoid of knots (under the connected sum operation) indexed by pairs of half integers. Doubly slice knots lie in the intersection of this bi-filtration. We construct examples of knots which illustrate the non-triviality of this bi-filtration at all levels. In particular, these are new examples of algebraically doubly slice knots that are not doubly slice, and many of these knots are slice. Cheeger-Gromov's von Neumann rho invariants play a key role to show non-triviality of this bi-filtration. We also show some classical invariants are reflected at the initial levels of this bi-filtration, and obtain a bi-filtration of the double concordance group.     

On the knot complement problem for non-hyperbolic knots

This paper explicitly provides two exhaustive and infinite families of pairs (M,k), where M is a lens space and k is a non-hyperbolic knot in M, which produces a manifold homeomorphic to M, by a non-trivial Dehn surgery. Then, we observe the uniqueness of such knot in such lens space, the uniqueness of the slope, and that there is no preserving homeomorphism between the initial and the final M's. We obtain further that Seifert fibered knots, except for the axes, and satellite knots are determined by their complements in lens spaces. An easy application shows that non-hyperbolic knots are determined by their complement in atoroidal and irreducible Seifert fibered 3-manifolds.     

On \pi-hyperbolic knots with the same 2-fold branched coverings

We consider the following problem from the Kirby's list (Problem 3.25): Let K be a knot in and M(K) its 2-fold branched covering space. Describe the equivalence class [K] of K in the set of knots under the equivalence relation if is homeomorphic to . It is known that there exist arbitrarily many different hyperbolic knots with the same 2-fold branched coverings, due to mutation along Conway spheres. Thus the most basic class of knots to investigate are knots which do not admit Conway spheres. In this paper we solve the above problem for knots which do not admit Conway spheres, in the following sense: we give upper bounds for the number of knots in the equivalence class [K] of a knot K and we describe how the different knots in the equivalence class of K are related. Received: 3 August 1998 / in final form: 17 June 1999     

Structure in the classical knot concordance group

We provide new information about the structure of the abelian group of topological concordance classes of knots in $S^3$. One consequence is that there is a subgroup of infinite rank consisting entirely of knots with vanishing Casson-Gordon invariants but whose non-triviality is detected by von Neumann signatures.     

A framization of the Hecke algebra of type B

We introduce a framization of the Hecke algebra of type B. For this framization, we construct a faithful tensorial representation and two linear bases. We also construct a Markov trace on such an algebra, and from this trace we derive isotopy invariants for framed and classical knots and links in the solid torus.     

Computing the Conway polynomial of several closures of oriented 3-braids

This paper deals with polynomial invariants of a class of oriented 3-string tangles and the knots (or links) obtained by applying six different closures. In Cabrera-Ibarra (2004) [1], expressions were given to compute the Conway polynomials of four different closures of the composition of two such 3-string tangles. By using the expressions and results from that reference, and using an algorithm developed on the basis of Giller?s calculations for 3-string tangles, we provide new results concerning six closures of 3-braids. Surprisingly, for 3-braids two of the closures turn out to be affine functions of the four previously defined. Among the contributions in this paper one finds computational tools to obtain the Conway polynomial of closures of 3-braids in terms of continuous fractions and their expansions. An interesting feature is that our calculations yield explicit, nonrecursive formulas in the case of 3-braids, thereby considerably lowering the time required to compute them. As a byproduct, explicit expressions are also given to obtain both numerators and denominators of continuous fractions in a nonrecursive way.     

Thin position for knots and 3-manifolds

We prove that for 2-bridge knots and 3-bridge knots in thin position the double branched cover inherits a manifold decomposition in thin position. We also argue that one should not expect this sort of correspondence to hold in general.