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"What you can tell about a link from the way it intersects a ball"; based on " An Obstruction to Embedding 4-Tangles in Links"
by
David A. Krebes
Calgary, Alberta CANADA
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. In a modern development, we discuss the sign of the fraction for an alternating tangle.
Paper reference: math.GT/9902119
Date received: March 9, 2011
Copyright © 2011 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cbcc-03.