STUDY ON KNOT INVARIANTS AND ITS APPLICATIONS

We studied on finite type invariants or Vassiliev invariants of ribbon 2-knots, HC-moves for ribbon 2-knots, some properties of HOMFLY polynomials of links, tangle surgeries preserving some polynomial invariants, and the finite type invariants for handcuff graphs. We defined finite type invariants for a class of ribbon 2-knots. Then we showed that each coefficient in the Taylor expansion of the normalized Alexander polynomial of a ribbon 2-knot is a Vassiliev invariant. There, we constructed a 'Vassiliev-like' filtration in two ways. However, we proved that the two filtrations are the same, and thus, the two finite type invariants are coincident. We defined the HC-move as an unknotting operation of a ribbon 2-knot as a generalization of a Δ-move for a 1-knot. Then we gave some relatins between the HC-move and the α_2-invariant of a ribbon 2-knot, which is the order 2 finite type invariant. This allowed us to decide the HC-unknotting numbers of some ribbon 2-konts. Making use of the virtual arc representation of a ribbon 2-knot due to Satoh, we saw that the HC-move corresponds to one of the "forbidden moves", which unknot every virtual knot. Then : (1) We proved that any virtural knot can be unknotted by the forbidden moves. (2) We proved the HC-move is an unknotting operation for the virtual arc representation of a ribbon 2-knot. (3) We gave some relation between the Δ-move for a 1-knot and the HC-move for the spun 2-knot. We give formulas for the second and third coefficient polynomials of the HOMFLY polynomial of a link which are described by the linking numbers and the coefficient polynomials of the HOMFLY polynomials of the proper sublinks. We introduce some tangle surgeries on the double of a tangle. If the tangle satisfies certain conditions, then the resulting link has the same polynomial invariant as the original one.