Volume 7, issue 1 (2007)

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Integrality of Homfly 1–tangle invariants

H R Morton

Algebraic & Geometric Topology 7 (2007) 327–338

arXiv: math.GT/0606336


Using spinning we analyze in a geometric way Haefliger’s smoothly knotted (4k1)–spheres in the 6k–sphere. Consider the 2–torus standardly embedded in the 3–sphere, which is further standardly embedded in the 6–sphere. At each point of the 2–torus we have the normal disk pair: a 4–dimensional disk and a 1–dimensional proper sub-disk. We consider an isotopy (deformation) of the normal 1–disk inside the normal 4–disk, by using a map from the 2–torus to the space of long knots in 4–space, first considered by Budney. We use this isotopy in a construction called spinning about a submanifold introduced by the first-named author. Our main observation is that the resultant spun knot provides a generator of the Haefliger knot group of knotted 3–spheres in the 6–sphere. Our argument uses an explicit construction of a Seifert surface for the spun knot and works also for higher-dimensional Haefliger knots.

Homfly, skein, annulus, quantum $sl(N)$, irreducible, integrality, 1–tangle
Mathematical Subject Classification 2000
Primary: 57M25, 57M27
Secondary: 57R56
Received: 18 December 2006
Accepted: 12 February 2007
Published: 29 March 2007
H R Morton
Department of Mathematical Sciences
University of Liverpool
Peach Street
Liverpool L69 7ZL