## Osaka Journal of Mathematics

### A note on knots with $\mathrm{H}(2)$-unknotting number one

Yuanyuan Bao

#### Abstract

We give an obstruction to unknotting a knot by adding a twisted band, derived from Heegaard Floer homology.

#### Article information

Source
Osaka J. Math., Volume 51, Number 3 (2014), 585-597.

Dates
First available in Project Euclid: 23 October 2014

https://projecteuclid.org/euclid.ojm/1414090792

Mathematical Reviews number (MathSciNet)
MR3272606

Zentralblatt MATH identifier
1304.62125

#### Citation

Bao, Yuanyuan. A note on knots with $\mathrm{H}(2)$-unknotting number one. Osaka J. Math. 51 (2014), no. 3, 585--597. https://projecteuclid.org/euclid.ojm/1414090792

#### References

• T. Abe, R. Hanaki and R. Higa: The unknotting number and band-unknotting number of a knot, Osaka J. Math. 49 (2012), no. 2, 523–550.
• B.E. Clark: Crosscaps and knots, Internat. J. Math. Math. Sci. 1 (1978), no. 1, 113–123.
• P. Gilmer and C. Livingston: The nonorientable four-genus of knots, J. London Math. Soc., doi:10.1112/jlms/jdr024 (2011).
• J. Hoste, Y. Nakanishi and K. Taniyama: Unknotting operations involving trivial tangles, Osaka J. Math. 27 (1990), 555–566.
• K. Ichihara and S. Mizushima: Crosscap numbers of pretzel knots, Topology Appl. 157 (2010), 193–201.
• T. Kanenobu and Y. Miyazawa: $H(2)$-unknotting number of a knot, Commun. Math. Res. 25 (2009), 433–460.
• W.B.R. Lickorish: Unknotting by adding a twisted band, Bull. London Math. Soc. 18 (1986), 613–615.
• \begingroup S.V. Matveev: Generalized surgeries of three-dimensional manifolds and representations of homology spheres, Mat. Zametki 42 (1987), 268–278, 345. \endgroup
• J.M. Montesinos: Surgery on links and double branched covers of $S^{3}$; in Knots, Groups, and $3$-Manifolds (Papers dedicated to the memory of R.H. Fox), Ann. of Math. Studies 84, Princeton Univ. Press, Princeton, NJ, 1975, 227–259.
• H. Murakami and Y. Nakanishi: On a certain move generating link-homology, Math. Ann. 284 (1989), 75–89.
• H. Murakami and A. Yasuhara: Four-genus and four-dimensional clasp number of a knot, Proc. Amer. Math. Soc. 128 (2000), 3693–3699.
• B. Owens: Unknotting information from Heegaard Floer homology, Adv. Math. 217 (2008), 2353–2376.
• P. Ozsváth and Z. Szabó: Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary, Adv. Math. 173 (2003), 179–261.
• P. Ozsváth and Z. Szabó: Holomorphic disks and three-manifold invariants: properties and applications, Ann. of Math. (2) 159 (2004), 1159–1245
• P. Ozsváth and Z. Szabó: Knots with unknotting number one and Heegaard Floer homology, Topology 44 (2005), 705–745.
• P. Ozsváth and Z. Szabó: On the Heegaard Floer homology of branched double-covers, Adv. Math. 194 (2005), 1–33.
• K. Taniyama and A. Yasuhara: On $C$-distance of knots, Kobe J. Math. 11 (1994), 117–127.
• A. Yasuhara: Connecting lemmas and representing homology classes of simply connected $4$-manifolds, Tokyo J. Math. 19 (1996), 245–261. \endthebibliography*