Satellite knots, to appear in Concise Encyclopedia of Knot Theory

Bridge numbers, to appear in Concise Encyclopedia of Knot Theory


Introduction to 3-manifolds, GSM 151, AMS, Providence, RI, 2014, x+286, ISBN: 978-1-4704-1020-9

Lecture notes on generalized Heegaard splittings, joint with T. Saito and M. Scharlemann, World Scientific, 2016, ISBN-13: 978-9813109117


The geometry of colors in van Gogh's Sunflowers, Heritage Science (2021) 9:136.

Surface complexes of Seifert fibered spaces, Proc. Amer. Math. Soc. 148 (2020) 3633--3645.

Kakimizu complexes of Seifert fibered spaces, Algebr. Geom. Topol. 18 (2018) no. 5, 2897--2918.

The Kakimizu complex of a surface, Topology Proc. 50 (2017) 111--139.

Contractibility of the Kakimizu complex and symmetric Seifert surfaces, with Piotr Przytycki, Trans. Amer. Math. Soc. 364 (2012) no. 3, 1489--1508.

Width complexes of knots and 3-manifolds, Pacific J. Math 239 (2009) no. 1, 135--156.

The Kakimizu complex is simply connected, with an appendix by Michael Kapovich, J. Topol 3 (2010) no. 4, 883--900.

Destabilizing amalgamated Heegaard splittings, with Richard Weidmann, Geometry & Topology Monographs Volume 12 (2007) 319--334.

Bridge numbers of torus knots, Math. Proc. Camb. Phil.Soc. 143 (2007) no. 3, 621--625.

Thin position of knots and 3-manifolds, with Hugh Howards, Topology Appl. 155 (2008) no. 13, 1371--1381.

Thin position of knots and 3-manifolds: A unified approach, with Hugh Howards, Yo'av Rieck, Geom. Topol. Monogr. 12. Geom. Topol. Publ. Coventry (2007) 89--120.

On the geometric and algebraic rank of graph manifolds, with Richard Weidmann, Pacific J. Math. 231 (2007) no. 2, 481--510.

Heegaard genus formula for Haken manifolds, Geom. Dedicata 119 (2006) 49--68.

3-manifolds with planar presentations and the width of satellite knots, with Marty Scharlemann, Trans. Amer. Math. Soc. 358 (2006) no. 9, 3781--3805.

Heegaard splittings of graph manifolds, Geometry & Topology (2004) 831--876.

Additivity of Bridge Numbers of Knots, Math. Proc. Camb. Phil. Soc 135 (2003) no. 3, 539--544.

Genus 2 hyperbolic 3-manifolds of arbitrarily large volume, Proc. Spring Topology and Dynamical Systems Conference (Morelia City, 2001). Topology Proc. 26 (2001/02) no. 1, 317--321.

Comparing Heegaard and JSJ structures or orientable 3-manifolds, with Marty Scharlemann, Trans. Amer. Math. Soc. 353 (2001) no. 2, 557--584.

Annuli in generalized Heegaard splittings and degeneration of tunnel number, with Marty Scharlemann, Math. Ann. 317 (2000) no. 4, 783--820.

The tunnel number of the sum of n knots is at least n, with Marty Scharlemann, Topology 38 (1999) no. 2, 265--270.

Tunnel numbers of small knots do not go down under connected sum, with Kanji Morimoto, Proc. Amer. Math. Soc. 128 (2000) no. 1, 269--278.

Additivity of tunnel number for small knots, Comment. Math. Helv. 75 (2000) no. 3, 353--367.

The stabilization problem for Heegaard splittings of Seifert fibered spaces, Topology Appl. 73 (1996) no. 2, 133--139.

Weakly reducible Heegaard splittings of Seifert fibered spaces, Topology Appl. 100 (2000) 2-3 219--222.

Irreducible Heegaard splittings of Seifert fibered spaces are either vertical or horizontal, with Yoav Moriah, Topology 37 (1998) no. 5, 1089--1112.

Heegaard splittings of Seifert fibered spaces with boundary Trans. Amer. Math. Soc. 347 (1995) no. 7, 2533--2552.

The classification of Heegaard splittings for (compact orientable surface) x circle Proc. LMS (3) 67 (1993) 425--448.

Rambling thoughts

Here are a few older manuscripts that are (at best) purely educational. I don't currently intend to publish these, for a variety of reasons. Of course, I could change my mind.

Waldhausen's "Heegaardzerlegungen der 3-Sphäre"

(This is an annotated and illustrated translation, written for the benefit of local graduate students. I recently noticed that S. Schleimer has something along these lines as well.)

Amalgamations of Heegaard splittings are unique

(This is addressed elsewhere, for instance by D. Bachman and M. Lackenby. The result seems kind of obvious, but there is an issue here.)