Symmetric chain complexes, twisted Blanchfield pairings and knot concordance

Two-solvable and two-bipolar knots with large four-genera

“…It is relatively easy to find examples of seed ribbon knots satisfying the hypotheses of Theorem 6.1, at least with the help of a computer program to compute twisted metabelian homology, as developed in [MP18]. In Section 6.1 we give one such example of a pair pR, αq, and describe the appropriate infection by knots with Arf invariant zero and large signature in order to obtain 2-solvable large 4-genus knots.…”

“…(More precisely, r1, 0s b rα`s and r1, 0s b rα´s generate.) As in [MP18], we computed the twisted homology using a Maple program, available for download on the authors' websites. The program obtains a presentation for the twisted homology using the Wirtinger presentation, taking the Fox derivatives, and then applying the representation.…”

“…In our method, it is also crucial to use L p2q -signatures over amenable groups that are not torsion-free, which were developed in [CO12,Cha14a] and deployed in a similar context in [MP18].…”

Twisted Blanchfield pairings and twisted signatures III: Applications

“…We briefly review the definition of twisted homology and some first facts about twisted Blanchfield forms; references include [2,14]. While we do not recall the definition of these pairings, we describe an algorithm to calculate them (due to Powell [15]) that may just as well be taken as a definition.…”

“…First, when β is induced by abelianisation, Bl β pKq reduces to the classical Blanchfield form BlpKq. Secondly, while Bl β pKq has been studied by Powell [15,16] and Miller-Powell [14], extracting concrete computable invariants from Bl β pKq has remained challenging. For instance, while Bl β pKq has been used to obstruct certain satellite knots from being slice [14], no explicit calculations have been possible for low crossing knots.…”