# Martin Scharlemann

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

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

[Abigail Thompson](/source/Abigail_Thompson) was a student of his.[2] Together they solved the [graph planarity](/source/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](/source/Mathematical_proof) of the classical theorem that [knots](/source/Knot_(mathematics)) with [unknotting number](/source/Unknotting_number) one are [prime](/source/Prime_knot). 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](/source/Topology_(journal))* **29** (1990), no. 4, 481–500.
- with [Abigail Thompson](/source/Abigail_Thompson), "Heegaard splittings of (surface) x I are standard" *[Mathematische Annalen](/source/Mathematische_Annalen)* **295** (1993), no. 3, 549–564.
- "Sutured manifolds and generalized Thurston norms", *[Journal of Differential Geometry](/source/Journal_of_Differential_Geometry)* **29** (1989), no. 3, 557–614.
- with [J. Hyam Rubinstein](/source/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](/source/Inventiones_mathematicae)* **82** (1985), no. 1, 37–55.
- with Maggy Tomova, "Alternate Heegaard genus bounds distance" *[Geometry & Topology](/source/Geometry_%26_Topology)* **10** (2006), 593–617.
- "Local detection of strongly irreducible Heegaard splittings" *[Topology and its Applications](/source/Topology_and_its_Applications)*, 1998
- with Abigail Thompson – "Link genus and the Conway moves" *[Commentarii Mathematici Helvetici](/source/Commentarii_Mathematici_Helvetici)*, 1989
- "Smooth spheres in \mathbb{R}^4 with four critical points are standard" *[Inventiones mathematicae](/source/Inventiones_mathematicae)*, 1985
- "Tunnel number one knots satisfy the Poenaru conjecture" *[Topology and its Applications](/source/Topology_and_its_Applications)*, 1984
- with A Thompson – "Detecting unknotted graphs in 3-space" *[Journal of Differential Geometry](/source/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"](http://www.math.ucsb.edu/~mgscharl/concise.html)

1. ["The Mathematics Genealogy Project – Martin Scharlemann"](http://www.genealogy.math.ndsu.nodak.edu/id.php?id=16934)

1. ["Geometric Topology in Dimensions 3 and 4"](http://users.math.yale.edu/~jj327/conference/)

1. ["2014 Class of the Fellows of the AMS"](https://www.ams.org/profession/ams-fellows/fellows2014.pdf). *Notices of the American Mathematical Society*. **61** (4): 420–421. April 2014.

1. "Detecting unknotted graphs in 3-space". *Journal of Differential Geometry*. **34** (2): 539–560. 1991. [doi:10.4310/jdg/1214447220](https://doi.org/10.4310/jdg/1214447220)

1. Lackenby, Marc (1997-08-01). "Surfaces, surgery and unknotting operations". *[Mathematische Annalen](/source/Mathematische_Annalen)*. **308** (4): 615–632. [doi:10.1007/s002080050093](https://doi.org/10.1007/s002080050093). [ISSN 0025-5831](https://www.worldcat.org/issn/0025-5831). [S2CID 121512073](https://api.semanticscholar.org/CorpusID:121512073)

1. Zhang, Xingru (1991-01-01). "Unknotting Number One Knots are Prime: A New Proof". *[Proceedings of the American Mathematical Society](/source/Proceedings_of_the_American_Mathematical_Society)*. **113** (2): 611–612. [doi:10.2307/2048550](https://doi.org/10.2307/2048550). [JSTOR 2048550](https://www.jstor.org/stable/2048550)

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