2015/09/30 by András Juhász, Andras Juhasz, Marco Marengon
Computer Science · Mathematics · #Floer homology #Geometric and Algebraic Topology #Homology (biology) #Homomorphism #Homotopy and Cohomology in Algebraic Topology #Knot (papermaking) #Knot complement #Knot invariant #Morphism #Seifert surface #Topological and Geometric Data Analysis #Tricolorability #math.GT #msc:57M27 #msc:57R58
paper · pdf · doi:10.2140/gt.2016.20.3623
published as Geom. Topol. 20 (2016) 3623-3673 · 38 pages, 3 figures, to appear in Geometry and Topology
openalex created_date 2016/06/24 · arxiv created 2016/08/09 · openalex publication_date 2016/12/21 · arxiv updated 2017/01/04 · openalex updated_date 2026/08/05
We show that a decorated knot concordance [math] from [math] to [math] induces a homomorphism [math] on knot Floer homology that preserves the Alexander and Maslov gradings. Furthermore, it induces a morphism of the spectral sequences to [math] that agrees with [math] on the [math] page and is the identity on the [math] page. It follows that [math] is nonvanishing on [math] . We also obtain an invariant of slice disks in homology 4–balls bounding [math] . ¶ If [math] is invertible, then [math] is injective, hence ¶\n<math display="block">\n<mrow> <mo class="qopname">dim</mo><msub><mrow><mover accent="false"><mrow><mo class="qopname">HFK</mo></mrow><mo class="qopname">̂</mo></mover></mrow><mrow><mi>j</mi></mrow></msub><mrow><mo class="MathClass-open">(</mo><mrow><mi>K</mi><mo class="MathClass-punc">,</mo><mi>i</mi></mrow><mo class="MathClass-close">)</mo></mrow> <mo class="MathClass-rel">≤</mo><mo class="qopname"> dim</mo><msub><mrow><mover accent="false"><mrow><mo class="qopname">HFK</mo></mrow><mo class="qopname">̂</mo></mover></mrow><mrow><mi>j</mi></mrow></msub><mrow><mo class="MathClass-open">(</mo><mrow><msup><mrow><mi>K</mi></mrow><mrow><mi>′</mi></mrow></msup><mo class="MathClass-punc">,</mo><mi>i</mi></mrow><mo class="MathClass-close">)</mo></mrow> </mrow>\n</math>\n¶ for every [math] . This implies an unpublished result of Ruberman that if there is an invertible concordance from the knot [math] to [math] , then [math] , where [math] denotes the Seifert genus. Furthermore, if [math] and [math] is fibred, then so is [math] .