Detecting knot invertibility

We discuss the consequences of the possibility that Vassiliev invariants do not detect knot invertibility as well as the fact that quantum Lie group invariants are known not to do so. On the other hand, finite group invariants, such as the set of homomorphisms from the knot group to M11, can detect knot invertibility. For many natural classes of knot invariants, including Vassiliev invariants and quantum Lie group invariants, we can conclude that the invariants either distinguish all oriented knots, or there exist prime, unoriented knots which they do not distinguish. Not long ago, two optimistic conjectures about the nature of Vassiliev invariants were widely circulated: Conjecture 1 Vassiliev invariants distinguish all prime, unoriented knots. Conjecture 2 [1] All Vassiliev invariants are linear combinations of derivatives at q = 1 of quantum Lie group invariants. Let s be a satellite operation on knots, so that if K is a knot, s(K) is a satellite of K. We say that an X-valued invariant V on knots intertwines s if V (s(K)) = sX(V (K)) for some sX : X → X, so that the invariant V ◦ s yields no more information than V itself. In this paper, we present the following three theorems. Theorem 3 Let V be a knot invariant which intertwines satellite operations. Then either V distinguishes all oriented knots, or there exist prime, unoriented knots which are not distinguished by V . More precisely, Theorem 3 states that, assuming V distinguishes prime, unoriented knots, it distinguishes any pair K1 and K2 of distinct oriented knots. Strictly speaking, we will only prove this when K2 is the inverse of K1 (i.e., K1 with the opposite orientation), but this is in fact the main case. The alternative case where they are not inverses is settled as follows: Knot inversion is itself a satellite operation, so V (K1) = V (K2) implies V (−K1) = V (−K2), which is to say that V does not distinguish K1 and K2 as unoriented knots. If they are not both prime, then by Lemma 7, they can be made prime by a satellite operation and they will remain indistinguishable. Theorem 4 The universal Vassiliev invariant vn of order n intertwines satellite operations. Theorem 5 The universal quantum link invariant far a Lie algebra g intertwines satellite operations.