Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 
 

 

Link diagrams with low Turaev genus


Author: Seungwon Kim
Journal: Proc. Amer. Math. Soc. 146 (2018), 875-890
MSC (2010): Primary 57M25
DOI: https://doi.org/10.1090/proc/13723
Published electronically: October 25, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We classify link diagrams with Turaev genus one and two in terms of an alternating tangle structure of the link diagram. The proof involves surgery along simple loops on the Turaev surface, called cutting loops, which have corresponding cutting arcs that are visible on the planar link diagram. These also provide new obstructions for a link diagram on a surface to come from the Turaev surface algorithm. We also show that inadequate Turaev genus one links are almost-alternating.


References [Enhancements On Off] (What's this?)

  • [1] Tetsuya Abe, The Turaev genus of an adequate knot, Topology Appl. 156 (2009), no. 17, 2704-2712. MR 2556029, https://doi.org/10.1016/j.topol.2009.07.020
  • [2] Tetsuya Abe and Kengo Kishimoto, The dealternating number and the alternation number of a closed 3-braid, J. Knot Theory Ramifications 19 (2010), no. 9, 1157-1181. MR 2726563, https://doi.org/10.1142/S0218216510008352
  • [3] Cody W. Armond and Adam M. Lowrance, Turaev genus and alternating decompositions, Algebr. Geom. Topol. 17 (2017), no. 2, 793-830. MR 3623672, https://doi.org/10.2140/agt.2017.17.793
  • [4] Cody Armond, Nathan Druivenga, and Thomas Kindred, Heegaard diagrams corresponding to Turaev surfaces, J. Knot Theory Ramifications 24 (2015), no. 4, 1550026, 14. MR 3346925, https://doi.org/10.1142/S0218216515500261
  • [5] Abhijit Champanerkar and Ilya Kofman, A survey on the Turaev genus of knots, Acta Math. Vietnam. 39 (2014), no. 4, 497-514. MR 3292579, https://doi.org/10.1007/s40306-014-0083-y
  • [6] Oliver T. Dasbach, David Futer, Efstratia Kalfagianni, Xiao-Song Lin, and Neal W. Stoltzfus, The Jones polynomial and graphs on surfaces, J. Combin. Theory Ser. B 98 (2008), no. 2, 384-399. MR 2389605, https://doi.org/10.1016/j.jctb.2007.08.003
  • [7] Chuichiro Hayashi, Links with alternating diagrams on closed surfaces of positive genus, Math. Proc. Cambridge Philos. Soc. 117 (1995), no. 1, 113-128. MR 1297898, https://doi.org/10.1017/S0305004100072947
  • [8] Adam M. Lowrance, Alternating distances of knots and links, Topology Appl. 182 (2015), 53-70. MR 3305610, https://doi.org/10.1016/j.topol.2014.12.010
  • [9] W. Menasco, Closed incompressible surfaces in alternating knot and link complements, Topology 23 (1984), no. 1, 37-44. MR 721450, https://doi.org/10.1016/0040-9383(84)90023-5
  • [10] Morwen B. Thistlethwaite, An upper bound for the breadth of the Jones polynomial, Math. Proc. Cambridge Philos. Soc. 103 (1988), no. 3, 451-456. MR 932668, https://doi.org/10.1017/S030500410006504X
  • [11] Tatsuya Tsukamoto, The almost alternating diagrams of the trivial knot, J. Topol. 2 (2009), no. 1, 77-104. MR 2499438, https://doi.org/10.1112/jtopol/jtp001
  • [12] Vladimir G. Turaev, A simple proof of the Murasugi and Kauffman theorems on alternating links, New Developments In The Theory of Knots. Series: Advanced Series in Mathematical Physics, ISBN: 978-981-02-0162-3. WORLD SCIENTIFIC, Edited by Toshitake Kohno, vol. 11, (1990), pp. 602-624. MR1087783

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 57M25

Retrieve articles in all journals with MSC (2010): 57M25


Additional Information

Seungwon Kim
Affiliation: Department of Mathematics, The Graduate Center, CUNY, 365 Fifth Avenue, New York, New York 10016
Email: skim2@gradcenter.cuny.edu

DOI: https://doi.org/10.1090/proc/13723
Received by editor(s): November 23, 2015
Received by editor(s) in revised form: October 2, 2016, and February 23, 2017
Published electronically: October 25, 2017
Communicated by: Kevin Whyte
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society