Milnor invariants and the HOMFLYPT polynomial

We give formulas expressing Milnor invariants of an n-component link L in the 3-sphere in terms of the HOMFLYPT polynomial as follows. If the Milnor invariant \bar{\mu}_J(L) vanishes for any sequence J with length at most k, then any Milnor \bar{\mu}-invariant \bar{\mu}_I(L) with length between 3 and 2k+1 can be represented as a combination of HOMFLYPT polynomial of knots obtained from the link by certain band sum operations. In particular, the `first non vanishing' Milnor invariants can be always represented as such a linear combination.Comment: Entirely revised version (20 pages). The main result was generalized and extended to Milnor invariants of links, using new arguments. Several corollaries are given, in particular one containing the main result of the previous version. Example and References adde

SH(3)-MOVE AND OTHER LOCAL MOVES ON KNOTS

An SH(3)-move is an unknotting operation on oriented knots introduced by Hoste, Nakanishi and Taniyama. We consider some relationships to other local moves such as a band surgery, Γ_0-move, and Δ-move, and give some criteria for estimating the SH(3)-unknotting number using the Jones, HOMFLYPT, Q polynomials. We also show a table of SH(3)-unknotting numbers for knots with up to 9 crossings

Unknotting numbers and triple point cancelling numbers of torus-covering knots

It is known that any surface knot can be transformed to an unknotted surface knot or a surface knot which has a diagram with no triple points by a finite number of 1-handle additions. The minimum number of such 1-handles is called the unknotting number or the triple point cancelling number, respectively. In this paper, we give upper bounds and lower bounds of unknotting numbers and triple point cancelling numbers of torus-covering knots, which are surface knots in the form of coverings over the standard torus $T$. Upper bounds are given by using $m$-charts on $T$ presenting torus-covering knots, and lower bounds are given by using quandle colorings and quandle cocycle invariants.Comment: 26 pages, 14 figures, added Corollary 1.7, to appear in J. Knot Theory Ramification

Finite type invariants of order 3 for a spatial handcuff graph

AbstractWe express a basis for the vector space of finite type invariants of order less than or equal to three for an embedded handcuff graph in a 3-sphere in terms of the linking number, the Conway polynomial, and the Jones polynomial of the sublinks of the handcuff graph