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We give a formula for the duality structure of the 3–manifold obtained by doing zero-framed surgery along a knot in the 3–sphere, starting from a diagram of the knot. We then use this to give a combinatorial algorithm for computing the twisted Blanchfield pairing of such 3–manifolds. With the twisting defined by Casson–Gordon-style representations, we use our computation of the twisted Blanchfield pairing to show that some subtle satellites of genus two ribbon knots yield nonslice knots. The construction is subtle in the sense that, once based, the infection curve lies in the second derived subgroup of the knot group.
Keywords: twisted Blanchfield pairing, symmetric Poincaré chain complex, knot concordance
Miller, Allison 1 ; Powell, Mark 2
@article{10_2140_agt_2018_18_3425,
author = {Miller, Allison and Powell, Mark},
title = {Symmetric chain complexes, twisted {Blanchfield} pairings and knot concordance},
journal = {Algebraic and Geometric Topology},
pages = {3425--3476},
publisher = {mathdoc},
volume = {18},
number = {6},
year = {2018},
doi = {10.2140/agt.2018.18.3425},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2018.18.3425/}
}
TY - JOUR AU - Miller, Allison AU - Powell, Mark TI - Symmetric chain complexes, twisted Blanchfield pairings and knot concordance JO - Algebraic and Geometric Topology PY - 2018 SP - 3425 EP - 3476 VL - 18 IS - 6 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/agt.2018.18.3425/ DO - 10.2140/agt.2018.18.3425 ID - 10_2140_agt_2018_18_3425 ER -
%0 Journal Article %A Miller, Allison %A Powell, Mark %T Symmetric chain complexes, twisted Blanchfield pairings and knot concordance %J Algebraic and Geometric Topology %D 2018 %P 3425-3476 %V 18 %N 6 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.2140/agt.2018.18.3425/ %R 10.2140/agt.2018.18.3425 %F 10_2140_agt_2018_18_3425
Miller, Allison; Powell, Mark. Symmetric chain complexes, twisted Blanchfield pairings and knot concordance. Algebraic and Geometric Topology, Tome 18 (2018) no. 6, pp. 3425-3476. doi: 10.2140/agt.2018.18.3425
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