2017/09/30 by Allison N. Miller, Mark Powell
Mathematics · Medicine · #Alexander polynomial #Botulinum Toxin and Related Neurological Disorders #Combinatorics #Fundamental group #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Knot (papermaking) #Knot complement #Knot invariant #Knot theory #Mathematics #Pairing #Physics #Pure mathematics #Trefoil knot #math.GT #msc:57M25 #msc:57M27 #msc:57N70
paper · pdf · doi:10.2140/agt.2018.18.3425
published as Algebr. Geom. Topol. 18 (2018) 3425-3476 · 38 pages, 7 figures. Version 2: revised discussion of examples of non-slice knots from the literature. Version 3: referee's comments incorporated, to be published in Algebraic and Geometric Topology
openalex created_date 2017/10/06 · arxiv created 2018/09/20 · openalex publication_date 2018/10/18 · arxiv updated 2018/10/24 · openalex updated_date 2026/08/05
We give a formula for the duality structure of the \n3 \n–manifold obtained by doing zero-framed surgery along a knot in the \n3 \n–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 \n3 \n–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.