Volume 12, Number 2, April 1985 | BY P. FREYD, D. YETTER; J. HOSTE; W. B. R. LICKORISH, K. MILLETT; AND A. OCNEANU
This paper announces a new isotopy invariant of oriented links in 3-space, represented by plane projections. The invariant, denoted as \( P \), is a function from the set of isotopy classes of tame oriented links to the set of homogeneous Laurent polynomials of degree 0 in \( x \), \( y \), and \( z \). It generalizes both the Alexander-Conway and Jones polynomials. The main theorem states that \( P \) satisfies the relation:
\[
xP_{L+}(x,y,z) + yP_{L-}(x,y,z) + zP_{L_0}(x,y,z) = 0,
\]
where \( L_+ \), \( L_0 \), and \( L_- \) are links with plane projections agreeing except in a small disk, and \( P_L(x,y,z) = 1 \) if \( L \) consists of a single unknotted component. The paper provides examples and discusses the properties of \( P \), including its behavior under Reidemeister moves and orientation changes. The authors also outline their approaches to defining and proving the existence of \( P \), including combinatorial and algebraic methods. The paper concludes with references to related work and acknowledges the contributions of Vaughan Jones.This paper announces a new isotopy invariant of oriented links in 3-space, represented by plane projections. The invariant, denoted as \( P \), is a function from the set of isotopy classes of tame oriented links to the set of homogeneous Laurent polynomials of degree 0 in \( x \), \( y \), and \( z \). It generalizes both the Alexander-Conway and Jones polynomials. The main theorem states that \( P \) satisfies the relation:
\[
xP_{L+}(x,y,z) + yP_{L-}(x,y,z) + zP_{L_0}(x,y,z) = 0,
\]
where \( L_+ \), \( L_0 \), and \( L_- \) are links with plane projections agreeing except in a small disk, and \( P_L(x,y,z) = 1 \) if \( L \) consists of a single unknotted component. The paper provides examples and discusses the properties of \( P \), including its behavior under Reidemeister moves and orientation changes. The authors also outline their approaches to defining and proving the existence of \( P \), including combinatorial and algebraic methods. The paper concludes with references to related work and acknowledges the contributions of Vaughan Jones.