Geodesic
Using secondary Upsilon invariants to rule out stable equivalence of knot complexes
Allen, Samantha1
1Department of Mathematics, Dartmouth College, Hanover, NH, United States
Algebraic and Geometric Topology, Tome 20 (2020) no. 1, pp. 29-48
Cet article a éte moissonné depuis la source Mathematical Sciences Publishers

Voir la notice de l'article

Two Heegaard Floer knot complexes are called stably equivalent if an acyclic complex can be added to each complex to make them filtered chain homotopy equivalent. Hom showed that if two knots are concordant, then their knot complexes are stably equivalent. Invariants of stable equivalence include the concordance invariants τ, 𝜀 and ϒ. Feller and Krcatovich gave a relationship between the Upsilon invariants of torus knots. We use secondary Upsilon invariants, defined by Kim and Livingston, to show that these relations do not extend to stable equivalence.

DOI : 10.2140/agt.2020.20.29
Classification : 57M25
Keywords: stable equivalence, Upsilon, torus knot

Allen, Samantha  1

1 Department of Mathematics, Dartmouth College, Hanover, NH, United States 
@article{10_2140_agt_2020_20_29,
     author = {Allen, Samantha},
     title = {Using secondary {Upsilon} invariants to rule out stable equivalence of knot complexes},
     journal = {Algebraic and Geometric Topology},
     pages = {29--48},
     year = {2020},
     volume = {20},
     number = {1},
     doi = {10.2140/agt.2020.20.29},
     url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2020.20.29/}
}
TY  - JOUR
AU  - Allen, Samantha
TI  - Using secondary Upsilon invariants to rule out stable equivalence of knot complexes
JO  - Algebraic and Geometric Topology
PY  - 2020
SP  - 29
EP  - 48
VL  - 20
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.2140/agt.2020.20.29/
DO  - 10.2140/agt.2020.20.29
ID  - 10_2140_agt_2020_20_29
ER  - 
%0 Journal Article
%A Allen, Samantha
%T Using secondary Upsilon invariants to rule out stable equivalence of knot complexes
%J Algebraic and Geometric Topology
%D 2020
%P 29-48
%V 20
%N 1
%U http://geodesic.mathdoc.fr/articles/10.2140/agt.2020.20.29/
%R 10.2140/agt.2020.20.29
%F 10_2140_agt_2020_20_29
Allen, Samantha. Using secondary Upsilon invariants to rule out stable equivalence of knot complexes. Algebraic and Geometric Topology, Tome 20 (2020) no. 1, pp. 29-48. doi: 10.2140/agt.2020.20.29

[1] M Borodzik, M Hedden, The Υ function of L–space knots is a Legendre transform, Math. Proc. Cambridge Philos. Soc. 164 (2018) 401 | DOI

[2] M Borodzik, C Livingston, Heegaard Floer homology and rational cuspidal curves, Forum Math. Sigma 2 (2014) | DOI

[3] P Feller, Optimal cobordisms between torus knots, Comm. Anal. Geom. 24 (2016) 993 | DOI

[4] P Feller, D Krcatovich, On cobordisms between knots, braid index, and the upsilon-invariant, Math. Ann. 369 (2017) 301 | DOI

[5] J Hom, The knot Floer complex and the smooth concordance group, Comment. Math. Helv. 89 (2014) 537 | DOI

[6] J Hom, A survey on Heegaard Floer homology and concordance, J. Knot Theory Ramifications 26 (2017) | DOI

[7] M H Kim, D Krcatovich, J Park, Links with non-trivial Alexander polynomial which are topologically concordant to the Hopf link, Trans. Amer. Math. Soc. 371 (2019) 5379 | DOI

[8] S G Kim, C Livingston, Secondary upsilon invariants of knots, Q. J. Math. 69 (2018) 799 | DOI

[9] R A Litherland, Signatures of iterated torus knots, from: "Topology of low-dimensional manifolds" (editor R A Fenn), Lecture Notes in Math. 722, Springer (1979) 71 | DOI

[10] C Livingston, Notes on the knot concordance invariant upsilon, Algebr. Geom. Topol. 17 (2017) 111 | DOI

[11] P S Ozsváth, A I Stipsicz, Z Szabó, Concordance homomorphisms from knot Floer homology, Adv. Math. 315 (2017) 366 | DOI

[12] P Ozsváth, Z Szabó, Knot Floer homology and the four-ball genus, Geom. Topol. 7 (2003) 615 | DOI

[13] P Ozsváth, Z Szabó, Holomorphic disks and knot invariants, Adv. Math. 186 (2004) 58 | DOI

[14] P Ozsváth, Z Szabó, On knot Floer homology and lens space surgeries, Topology 44 (2005) 1281 | DOI

Cité par Sources :