Twisted Blanchfield pairings and twisted signatures I: Algebraic background

Nonslice linear combinations of iterated torus knots

“…The result of [14] is proved without having to study invariant metabolisers; see also [5,Section 6]. This is a feature of iterated torus knots T .p; Q/ with p D 2 and fails when p > 2.…”

“…For each given Z p -invariant metaboliser M of p .J /, the "sliceness-obstructing character" that we will produce will be of the form D 1 ˚ 2 ˚ ˚ , where  denotes the trivial character. Using the definition of J, together with the direct sum decomposition of [4,Corollary 4.21], the Witt class of the metabelian Blanchfield pairing of J is given by (2) Bl ˛.p; / .J / Bl ˛.p; 1 / .T .p; qI p; q 1 // ˚ Bl ˛.p; 2 / .T .p; q 1 // ˚ Bl ˛.p; / .T .p; qI p; q 2 // ˚Bl ˛.p; / .T .p; q 2 //: This expression can be further decomposed by applying the satellite formula for the metabelian Blanchfield forms given in [4,Theorem 4.19]. Regardless of the final expression, the problem has been converted into a question of linear independence in W .C.t /; COEt ˙1/.…”

“…We use ˛.p; / WD ˛K.p; / to denote the metabelian representation that was described in Section 3.1. The behaviour of metabelian Blanchfield pairings under connected sums [4,Corollary 4.21] implies that Bl ˛.p; / .K/ is Witt equivalent to the linking form (23)…”

“…For a sequence S D .q 1 ; : : : ; q k /, we use T .p; y S / to denote the iterated torus knot T .p; q 1 I : : : I p; q k 1 /. Next, we apply the satellite formula for the metabelian Blanchfield pairing [4,Theorem 4.19] to both expressions in (23). As we are working with pfold covers and the sequences Q 2i 1 and Q 2i (resp.…”