An Obstruction to Embedding 4-Tangles in Links

Details An Obstruction to Embedding 4-Tangles in Links An Obstruction to Embedding 4-Tangles in Links

1999-02-21

arxiv.org/abs/math/9902119v1

Mathematics

Geometric Topology

32 pages, 17 figures. To appear in the Journal of Knot Theory and its Ramifications Vol. 8 No. 3 (May 99)

Scientific paper

We consider the ways in which a 4-tangle T inside a unit cube can be extended outside the cube into a knot or link L. We present two links n(T) and d(T) such that the greatest common divisor of the determinants of these two links always divides the determinant of the link L. In order to prove this result we give a two-integer invariant of 4-tangles. Calculations are facilitated by viewing the determinant as the Kauffman bracket at a fourth root of -1, which sets the loop factor to zero. For rational tangles, our invariant coincides with the value of the associated continued fraction.

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