Martin Scharlemann

Martin George Scharlemann (born 6 December 1948) is an American topologist who is a professor at the University of California, Santa Barbara. ^[1] He obtained his Ph.D. from the University of California, Berkeley under the guidance of Robion Kirby in 1974. ^[2]

A conference in his honor was held in 2009 at the University of California, Davis. ^[3] He is a Fellow of the American Mathematical Society, for his "contributions to low-dimensional topology and knot theory." ^[4]

Abigail Thompson was a student of his. ^[2] Together they solved the graph planarity problem: There is an algorithm to decide whether a finite graph in 3-space can be moved in 3-space into a plane. ^[5]

He gave the first proof of the classical theorem that knots with unknotting number one are prime. He used hard combinatorial arguments for this. Simpler proofs are now known. ^{[6] [7]}

Selected publications

• "Producing reducible 3-manifolds by surgery on a knot" Topology 29 (1990), no. 4, 481–500.
• with Abigail Thompson, "Heegaard splittings of (surface) x I are standard" Mathematische Annalen 295 (1993), no. 3, 549–564.
• "Sutured manifolds and generalized Thurston norms", Journal of Differential Geometry 29 (1989), no. 3, 557–614.
• with J. Hyam Rubinstein, "Comparing Heegaard splittings of non-Haken 3-manifolds" Topology35 (1996), no. 4, 1005–1026
• "Unknotting number one knots are prime", Inventiones mathematicae 82 (1985), no. 1, 37–55.
• with Maggy Tomova, "Alternate Heegaard genus bounds distance" Geometry & Topology 10 (2006), 593–617.
• "Local detection of strongly irreducible Heegaard splittings" Topology and its Applications , 1998
• with Abigail Thompson – "Link genus and the Conway moves" Commentarii Mathematici Helvetici , 1989
• "Smooth spheres in ${\displaystyle \mathbb {R} ^{4}}$ with four critical points are standard" Inventiones mathematicae , 1985
• "Tunnel number one knots satisfy the Poenaru conjecture" Topology and its Applications , 1984
• with A Thompson – "Detecting unknotted graphs in 3-space" Journal of Differential Geometry , 1991
• with A Thompson – "Thin position and Heegaard splittings of the 3-sphere" J. Differential Geom, 1994

References

1. "Curriculum Vitae – Martin Scharlemann".
2. "The Mathematics Genealogy Project – Martin Scharlemann".
3. "Geometric Topology in Dimensions 3 and 4".
4. "2014 Class of the Fellows of the AMS" (PDF). Notices of the American Mathematical Society. 61 (4): 420–421. April 2014.
5. Scharlemann, Martin; Thompson, Abigail (1991). "Detecting unknotted graphs in 3-space". Journal of Differential Geometry. 34 (2): 539–560. doi: 10.4310/jdg/1214447220 .
6. Lackenby, Marc (1997-08-01). "Surfaces, surgery and unknotting operations". Mathematische Annalen . 308 (4): 615–632. doi:10.1007/s002080050093. ISSN   0025-5831. S2CID   121512073.
7. Zhang, Xingru (1991-01-01). "Unknotting Number One Knots are Prime: A New Proof". Proceedings of the American Mathematical Society . 113 (2): 611–612. doi: 10.2307/2048550 . JSTOR   2048550.