## Algebraic & Geometric Topology

### Slice knots which bound punctured Klein bottles

Arunima Ray

#### Abstract

We investigate the properties of knots in $S3$ which bound punctured Klein bottles, such that a pushoff of the knot has zero linking number with the knot, ie has zero framing. This is motivated by the many results in the literature regarding slice knots of genus one, for example, the existence of homologically essential zero self-linking simple closed curves on genus one Seifert surfaces for algebraically slice knots. Given a knot $K$ bounding a punctured Klein bottle $F$ with zero framing, we show that $J$, the core of the orientation preserving band in any disk–band form of $F$, has zero self-linking. We prove that such a $K$ is slice in a $ℤ[1∕2]$–homology $B4$ if and only if $J$ is as well, a stronger result than what is currently known for genus one slice knots. As an application, we prove that given knots $K$ and $J$ and any odd integer $p$, the $(2,p)$–cables of $K$ and $J$ are $ℤ[1∕2]$–concordant if and only if $K$ and $J$ are $ℤ[1∕2]$–concordant. In particular, if the $(2,1)$–cable of a knot $K$ is slice, $K$ is slice in a $ℤ[1∕2]$–homology ball.

#### Article information

Source
Algebr. Geom. Topol., Volume 13, Number 5 (2013), 2713-2731.

Dates
Revised: 15 March 2013
Accepted: 17 March 2013
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715688

Digital Object Identifier
doi:10.2140/agt.2013.13.2713

Mathematical Reviews number (MathSciNet)
MR3116301

Zentralblatt MATH identifier
1271.57027

Keywords
knot concordance

#### Citation

Ray, Arunima. Slice knots which bound punctured Klein bottles. Algebr. Geom. Topol. 13 (2013), no. 5, 2713--2731. doi:10.2140/agt.2013.13.2713. https://projecteuclid.org/euclid.agt/1513715688

#### References

• J C Cha, The structure of the rational concordance group of knots, Mem. Amer. Math. Soc. 189 (2007) 95
• J C Cha, C Livingston, D Ruberman, Algebraic and Heegaard–Floer invariants of knots with slice Bing doubles, Math. Proc. Cambridge Philos. Soc. 144 (2008) 403–410
• B E Clark, Crosscaps and knots, Internat. J. Math. Math. Sci. 1 (1978) 113–123
• T D Cochran, C D Davis, Counterexamples to Kauffman's conjectures on slice knots
• T D Cochran, C D Davis, A Ray, Injectivity of satellite operators in knot concordance, to appear in Journal of Topology
• T D Cochran, B D Franklin, M Hedden, P D Horn, Knot concordance and homology cobordism, Proc. Amer. Math. Soc. 141 (2013) 2193–2208
• T D Cochran, S Friedl, P Teichner, New constructions of slice links, Comment. Math. Helv. 84 (2009) 617–638
• T D Cochran, S Harvey, C Leidy, Derivatives of knots and second-order signatures, Algebr. Geom. Topol. 10 (2010) 739–787
• T D Cochran, K E Orr, Not all links are concordant to boundary links, Ann. of Math. 138 (1993) 519–554
• T D Cochran, K E Orr, P Teichner, Knot concordance, Whitney towers and $L\sp 2$–signatures, Ann. of Math. 157 (2003) 433–519
• T D Cochran, K E Orr, P Teichner, Structure in the classical knot concordance group, Comment. Math. Helv. 79 (2004) 105–123
• D Cooper, Signatures of surfaces with applications to knot and link cobordism, PhD thesis, University of Warwick (1982)
• P M Gilmer, Slice knots in $S\sp{3}$, Quart. J. Math. Oxford Ser. 34 (1983) 305–322
• P M Gilmer, Classical knot and link concordance, Comment. Math. Helv. 68 (1993) 1–19
• P M Gilmer, C Livingston, On surgery curves for genus one slice knots, to appear in Pacific Journal of Mathematics
• C M Gordon, R A Litherland, On the signature of a link, Invent. Math. 47 (1978) 53–69
• M Hedden, P Kirk, Instantons, concordance, and Whitehead doubling, J. Differential Geom. 91 (2012) 281–319
• M Hedden, C Livingston, D Ruberman, Topologically slice knots with nontrivial Alexander polynomial, Adv. Math. 231 (2012) 913–939
• M Hirasawa, M Teragaito, Crosscap numbers of $2$–bridge knots, Topology 45 (2006) 513–530
• K Ichihara, S Mizushima, Crosscap numbers of pretzel knots, Topology Appl. 157 (2010) 193–201
• L H Kauffman, On knots, Annals of Mathematics Studies 115, Princeton Univ. Press (1987)
• A Kawauchi, Rational-slice knots via strongly negative-amphicheiral knots, Commun. Math. Res. 25 (2009) 177–192
• C Kearton, Cobordism of knots and Blanchfield duality, J. London Math. Soc. 10 (1975) 406–408
• R Kirby, $4$–manifold problems, from: “Four-manifold theory”, (C Gordon, R Kirby, editors), Contemp. Math. 35, Amer. Math. Soc. (1984) 513–528
• R Kirby, Problems in low-dimensional topology, from: “Geometric topology”, (W H Kazez, editor), AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc. (1997) 35–473
• J Levine, Invariants of knot cobordism, Invent. Math. 8 (1969) 98–110 addendum in Invent. Math. 8 (1969) 355
• J Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969) 229–244
• W B R Lickorish, An introduction to knot theory, Graduate Texts in Mathematics 175, Springer, New York (1997)
• R A Litherland, Signatures of iterated torus knots, from: “Topology of low-dimensional manifolds”, (R A Fenn, editor), Lecture Notes in Math. 722, Springer, Berlin (1979) 71–84
• C Livingston, P Melvin, Abelian invariants of satellite knots, from: “Geometry and topology”, (J Alexander, J Harer, editors), Lecture Notes in Math. 1167, Springer, Berlin (1985) 217–227
• W H Meeks, III, Representing codimension-one homology classes on closed nonorientable manifolds by submanifolds, Illinois J. Math. 23 (1979) 199–210
• H Murakami, A Yasuhara, Crosscap number of a knot, Pacific J. Math. 171 (1995) 261–273
• K Murasugi, The Arf invariant for knot types, Proc. Amer. Math. Soc. 21 (1969) 69–72
• P Ozsváth, Z Szabó, Knot Floer homology and the four-ball genus, Geom. Topol. 7 (2003) 615–639
• T M Price, Homeomorphisms of quaternion space and projective planes in four space, J. Austral. Math. Soc. Ser. A 23 (1977) 112–128
• J A Rasmussen, Floer homology and knot complements, PhD thesis, Harvard University (2003) Available at \setbox0\makeatletter\@url http://search.proquest.com/docview/305332635 {\unhbox0
• L P Roberts, Some bounds for the knot Floer $\tau$–invariant of satellite knots, Algebr. Geom. Topol. 12 (2012) 449–467
• D Rolfsen, Knots and links, Mathematics Lecture Series 7, Publish or Perish, Houston, TX (1990)
• M Teragaito, Creating Klein bottles by surgery on knots, J. Knot Theory Ramifications 10 (2001) 781–794
• M Teragaito, Crosscap numbers of torus knots, Topology Appl. 138 (2004) 219–238