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" Topology 35 (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 R 4 \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
"Curriculum Vitae – Martin Scharlemann".
"The Mathematics Genealogy Project – Martin Scharlemann".
"Geometric Topology in Dimensions 3 and 4".
https://www.ams.org/profession/ams-fellows/fellows2014.pdf[bare URL PDF]
Scharlemann, Martin; Thompson, Abigail (1991). "Detecting unknotted graphs in 3-space". Journal of Differential Geometry. 34 (2): 539–560. doi:10.4310/jdg/1214447220.
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.
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.
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