A new knot invariant

Abstract

Recall that a torus knot is an honest knot (not isotopic to a circle in a plane) which lies in an unknot ted embedded torus M c I R 3. An unknot ted torus M has two embedded oriented circles ~ and q, representing homology classes 14[ and Ir/I of a preferred basis for the homology group over 7/, Ha (M), namely bounding in the interior and in the exterior o f M c l g 3 respectively. If 7:S1--~M~]R 3 is a torus knot then the absolute values o f the intersection numbers in Ha(M) of 7 with ~ and r/, are denoted p and q respectively and one can assume 2 < p < q. We may have to modify M (exchange inside and outside) wi thout moving 7, to see this. The integers p and q are coprime and they characterize the isotopy class of the torus knot completely but for reflection in a plane. There is the s tandard model ~p,q for t hep q t o r u s knot defined in Sect. 2, (2.2). For suitable M, ~, r/, 7, the intersection numbers are [~ l~ l r / [= I r / [ c~ ]~ l= l ~H0(M) , [7[ c-~l~l =p , tTl~l~l=q, and I71=ql~l-pl~/[.