Journal of Knot Theory and Its Ramifications

In this paper, we study on knots and closed incompressible surfaces in the 3-sphere via Morse functions. We show that both knots and closed incompressible surfaces can be isotoped into a "related Morse position" simultaneously. As an application, we have the following results.

• Smallness of Montesinos tangles with length two and Kinoshita's theta curve.

• Classification of closed incompressible and meridionally incompressible surfaces in 2-bridge theta-curve and handcuff graph complements and the complements of links which admit Hopf tangle decompositions.

In this paper, we study the arithmetic of the minimal splitting field of the Alexander polynomial of a knot and present two kinds of infinite families of knots, one being a family of knots which satisfy Heilbronn conjecture and the other a family of counterexamples to the conjecture.

We use Reidemeister torsion to study a twisted Alexander polynomial, as defined by Turaev, for links in the projective space. Using sign-refined torsion, we derive a skein relation for a normalized form of this polynomial.

Using Yoshikawa's surface diagram, we constructed new invariants of ambient isotopy classes of smoothly embedded closed surfaces in ℝ⁴ via a state-sum model similar to the Kauffman's state-sum model for the Jones polynomial for classical knots and links in ℝ³. It is shown that the invariants can also be defined by skein relation and thus they are calculated from Yoshikawa's surface diagrams recurrently. Some of the properties of the invariants are given and explicit computations for several surfaces are included.

In this paper, we consider the minimal genus problem in a ruled 4-manifold M. There are three key ingredients in the studying, the action of diffeomorphism group of M on H²(M,Z), the geometric construction of surfaces representing a cohomology class and the generalized adjunction formula. At first, we discuss the standard form (see Definition 1.1) of a class under the action of diffeomorphism group on H²(M,Z), we prove the uniqueness of the standard form. Then we construct some embedded surfaces representing the standard forms of some positive classes, the generalized adjunction formula is used to show that these surfaces realize the minimal genera.

Whether Vassiliev invariants can distinguish all knots or not is a well-known open problem which is equivalent to the question whether the similarity index of any two different knots is finite or not.

Let T and S be two tangles which are n-similar for some natural number n and let the closure of T be well-defined. Let T* and S* be the mirror images of T and S respectively. Then we show that for any prime number p, mod p for any integral Vassiliev invariant v of degree ≤ np. We also show that for any Vassiliev invariant w of degree ≤ n if n is odd. Therefore, if an amphicheiral knot can be distinguished from a trivial knot by a Vassiliev invariant, then it has an even triviality index.

From these, we get some necessary conditions for a knot invariant to be a Vassiliev invariant and get a method to detect the similarity index of two knots or tangles.

The colored Jones function of a knot is a sequence of Laurent polynomials. It was shown by Le and the author that such sequences are q-holonomic, that is, they satisfy linear q-difference equations with coefficients Laurent polynomials in q and qⁿ. We show from first principles that q-holonomic sequences give rise to modules over a q-Weyl ring. Frohman–Gelca–LoFaro have identified the latter ring with the ring of even functions of the quantum torus, and with the Kauffman bracket skein module of the torus. Via this identification, we study relations among the orthogonal, peripheral and recursion ideal of the colored Jones function, introduced by the above mentioned authors. In the second part of the paper, we convert the linear q-difference equations of the colored Jones function in terms of a hierarchy of linear ordinary differential equations for its loop expansion. This conversion is a version of the WKB method, and may shed some information on the problem of asymptotics of the colored Jones function of a knot.

This paper summarizes substantive new results derived by a student team (the first three authors) under the direction of the fourth author at the 2005 session of the KSU REU "Brainstorming and Barnstorming". The main results are a decomposition theorem for quandles in terms of an operation of "semidisjoint union" showing that all finite quandles canonically decompose via iterated semidisjoint unions into connected subquandles, and a structure theorem for finite connected quandles with prescribed inner automorphism group. The latter theorem suggests a new approach to the classification of finite connected quandles.