Transactions AMS **349** (1997) 4429-4464

Here's a picture of a *twisting *of a knot:

If you'd like to see a "magic trick" involving this operation, just click. (Due to K. Motegi)

Y. Mathieu invented this twisting operation in 1989 or so, and asked
which knots are related by twisting. A twisting of a knot *K *is parametrized
by a choice of oriented twisting torus *V *and number of full twists
*d *(=delta in paper). In 1991 Motegi conjectured that each presentation
of *K *in *V *gives rise to at most one composite twisted unknot
and that this composite knot arises from a single twist of *K *. In
that paper he proved that the number of twists *d *must be less than
six.

Miyazaki next gave a necessary condition for a composite knot to be a twisted unknot, ruling out, for example, the granny knot; about then Teragaito gave some especially amusing examples of composite twisted unknots.

In 1993 I proved the following theorems, adapting the techniques of John Luecke and Cameron Gordon:

**1. If an unknot is given more than one full twist, the result is prime.
**

That is, if *K *is a composite twisted unknot, d=1.

**2. If a composite knot is given three (non-trivial) full twists, the
result is prime. If a composite knot is given two (non-trivial) full twists
and the result is composite, we began with a granny knot and obtained the
granny knot of opposite handedness. **

The granny knot case is due to Motegi and Hayashi, and required very special attention in the proof (an additional strand had to be woven through the induction. From a very technical standpoint, this is the most interesting part of the paper.)

**Corollary to 2. **Thus, if twisting an unknot one full twist in
one direction gives a composite knot, twisting one full twist in the other
direction gives a prime knot. That is, given an unknot and twisting torus
V, there's at most one twisting of V that gives a composite knot.

So much for negative results. Here are the positive results:

**3 : If K is a composite twisted unknot such that d=1 and (*),
then K in V(1) is of the form K =S(d) #-k_1 , where**

**(i) k is a torus 1-bridge knot with presentation in and on a solid
torus V';
(ii) -k_1 is the reflection of a (1,V')-twisting of k;
(iii) and S(k) is any restricted band sum with respect to V' of k with a
collection of disjoint (1,1) and (0,0) curves on the boundary of V' .**

**Furthermore, all knots K=S(k)#(-k_1) , as described, are indeed composite
twisted unknots.**

(1-bridge torus knot and restricted band sum are defined in the paper)

where (*) is the condition that the intersection of the splitting sphere of K and the twisting torus V has two components. I give several examples that show that the techniques used so far break down completely, in several ways, when approaching (*). Since Theorem 1 assures d must equal 1, the classification is thus complete, up to this renegade hypothesis.

**Corollary to 3 **All known examples, although seeming quite different,
are easily given a representation in the form outlined. The picture above
is a graphical shorthand for one of Teragaito's examples, put in the form
of Theorem 3.

**4 Finally, for fun: Given any composite knot K and any (j,2)
torus knot there are an infinite number of twisting tori around K that
yield, after a single twist, a non-trivial composite knot with one summand
the (j,2) torus knot. **

Independently, Motegi and Hayashi proved Theorem 1; Teragaito proved a weaker version of Theorem 1; and Motegi and Hayashi proved the first sentence in Theorem 2 and gave the very amusing granny knot example mentioned in that theorem.

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