On Composite Twisted Unknots

Following Mathieu [Ma], Motegi [Mo] and others, we consider the class of possible composite twisted unknots as well as pairs of composite knots related by twisting. At most one composite knot can arise from a particular V -twisting of an unknot; moreover a twisting of the unknot cannot be composite if we have applied more than a single full twist. A pair of composite knots can be related through at most one full twist for a particular V -twisting, or one summand was una ected by the twist, or the knots were the right and left handed granny knots. Finally a conjectured characterization of all composite twisted unknots that do arise is given. Figure 1: A twisting of a knot Following Mathieu [Ma], Motegi [Mo] and others, we consider the class of possible composite twisted unknots. In essence, a twisting of a given knot K is constructed by pulling several strands away from the knot, momentarily cutting them, giving them a few full twists, and rejoining the strands to each other as they were before, as illustrated in gure 1. A knot is composite if it 1 can be described as two simpler knots spliced together. Motegi conjectured, and we show in Section 3, that a twisting of the unknot cannot be composite if we have given the unknot more than a single twist. In Section 2, we also describe all known examples of composite knots that do arise from a single twist of the unknot, and conjecture the list is complete. But once we have chosen our strands to twist, can we obtain two composite twisted unknots, the rst from twisting once one way, the other from twisting once in the other direction? In Section 4, we show that a pair of composite knots can be related through at most a single full twist, or the twist a ected only one summand, or that the knots were the right and left handed granny knots, which cannot be twisted unknots by [MY]. The pair of composite knots arising from the same twisting of an unknot would be twistings of one another, related through two full twists, an impossibility, since the only such knots cannot be twisted unknots. Finally we show every composite knot is related to an in nite number of other composite knots through some full twist. Let us formalize our terms. Let K be a knot in the interior of a standardly embedded solid torus V in S ( gure 1). Let ; @V be the standard longitude, meridian of V in S. For a simple closed curve @V such that [ ] = [ ] + [ ] in H1(@V ; Z), let V ( ) = V ( ) = V [ (solid torus such that meridian lies on ) = S. We assume that the minimal geometric intersection of K with a meridinal disk of V is at least two. K = K V ( ) then depends on K in V and 2 Z. Generally we assume > 0, by our choice of orientation of . If K is unknotted in V (0) = S, we call K a twisted unknot. If K is a composite knot in V (0) we call K a twisted composite knot. In general we consider K to be a ( ; V )-twisting of K. We occasionally refer to a V -twisting, a ( ; V )-twisting for any . Mathieu [Ma] described this twisting operation. A knot K is composite if a splitting sphere can separate S into balls each containing a single, knotted arc of K, a knot summand. In our analysis of composite twisted unknots that arise when = 1, there are two cases: either there is a sphere splitting the knot summands that is punctured twice by @V or every splitting sphere must be punctured at least four times. With Theorem 2.2 we classify all composite twisted unknots arising from the former.