In [Towards invariants of surfaces in 4-space via classical link invariants, Trans. Amer. Math. Soc.361 (2009) 237–265], Lee defined a polynomial [[D]] for marked graph diagrams D of surface-links in 4-space by using a state-sum model involving a given classical link invariant. In this paper, we deal with some obstructions to obtain an invariant for surface-links represented by marked graph diagrams D by using the polynomial [[D]] and introduce an ideal coset invariant for surface-links, which is defined to be the coset of the polynomial [[D]] in a quotient ring of a certain polynomial ring modulo some ideal and represented by a unique normal form, i.e. a unique representative for the coset of [[D]] that can be calculated from [[D]] with the help of a Gröbner basis package on computer.

A marked graph diagram is a link diagram possibly with marked 4-valent vertices. S. J. Lomonaco, Jr. and K. Yoshikawa introduced a method of representing surface-links by marked graph diagrams. Specially, K. Yoshikawa suggested local moves on marked graph diagrams, nowadays called Yoshikawa moves. It is now known that two marked graph diagrams representing equivalent surface-links are related by a finite sequence of these Yoshikawa moves. In this paper, we provide some generating sets of Yoshikawa moves on marked graph diagrams representing unoriented surface-links, and also oriented surfacelinks. We also discuss independence of certain Yoshikawa moves from the other moves.Mathematics Subject Classification 2000: 57Q45; 57M25.

We introduce a polynomial invariant of virtual links that is non-trivial for many virtuals, but is trivial on classical links. Also this polynomial is sometimes useful to find the virtual crossing number of virtual knots. We give various properties of this polynomial and examples.

Abstract. In this paper, we introduce a method to construct ambient isotopy invariants for smooth imbeddings of closed surfaces into 4-space by using hyperbolic splittings of the imbedded surfaces and an arbitrary given isotopy or regular isotopy invariant of classical knots and links in 3-space. Using this construction, adopting the Kauffman bracket polynomial as an example, we produce some invariants.

We first report on the fabrication of 2 inch wafer-level GaN-based vertical light-emitting diodes (LEDs) by using a multi-functional bonding material system, which is composed of a thick Cu diffusion barrier and a bonding layer. The bonding material system superbly absorbs laser-induced stress and also effectively serves as a barrier to the indiffusion of Sn to the active region. Fully packaged vertical LEDs fabricated with indium tin oxide (ITO)/AgCu contact and the bonding material system give an operating voltage of 3.35 V at 350 mA. After over 1800 h, the operating voltages remain stable, and the reverse currents are in the range 3-8 × 10 −7 A at −5 V.