Download this article
 Download this article For screen
For printing
Recent Issues

Volume 26
Issue 6, 1965–2352
Issue 5, 1597–1963
Issue 4, 1229–1596
Issue 3, 825–1227
Issue 2, 411–824
Issue 1, 1–410

Volume 25, 9 issues

Volume 24, 9 issues

Volume 23, 9 issues

Volume 22, 8 issues

Volume 21, 7 issues

Volume 20, 7 issues

Volume 19, 7 issues

Volume 18, 7 issues

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
 
Subscriptions
 
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
 
Author index
To appear
 
Other MSP journals
A structural approach to tree decompositions of knots and spatial graphs

Corentin Lunel and Arnaud de Mesmay

Algebraic & Geometric Topology 26 (2026) 2015–2047
Bibliography
1 C C Adams, The knot book: an elementary introduction to the mathematical theory of knots, Amer. Math. Soc. (2004) MR2079925
2 O Amini, F Mazoit, N Nisse, S Thomassé, Submodular partition functions, Discrete Math. 309 (2009) 6000 MR2552632
3 R H Bing, The geometric topology of 3-manifolds, 40, Amer. Math. Soc. (1983) MR728227
4 R Blair, M Ozawa, Height, trunk and representativity of knots, J. Math. Soc. Japan 71 (2019) 1105 MR4023298
5 H L Bodlaender, B Burton, F V Fomin, A Grigoriev, Knot diagrams of treewidth two, from: "Graph-theoretic concepts in computer science" (editors I Adler, H Müller), Lecture Notes in Comput. Sci. 12301, Springer (2020) 80 MR4184208
6 B A Burton, The HOMFLY-PT polynomial is fixed-parameter tractable, from: "34th International Symposium on Computational Geometry" (editors B Speckmann, C D Tóth), LIPIcs. Leibniz Int. Proc. Inform. 99, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern (2018) 18 MR3824262
7 B Burton, H Edelsbrunner, J Erickson, S Tillmann, Computational geometric and algebraic topology, Oberwolfach Rep. 12 (2015) 2637 MR3586035
8 B A Burton, C Maria, J Spreer, Algorithms and complexity for Turaev–Viro invariants, J. Appl. Comput. Topol. 2 (2018) 33 MR3873178
9 R Diestel, Graph theory, 173, Springer (2017) MR3644391
10 R Diestel, S i Oum, Tangle-tree duality : in graphs, matroids and beyond, Combinatorica 39 (2019) 879 MR4015355
11 R Diestel, S i Oum, Tangle-tree duality in abstract separation systems, Adv. Math. 377 (2021) 107470 MR4186013
12 R Diestel, G Whittle, Tangles and the Mona Lisa, (2016) arXiv:1603.06652
13 F Dorn, E Penninkx, H L Bodlaender, F V Fomin, Efficient exact algorithms on planar graphs : exploiting sphere cut branch decompositions, from: "Algorithms—ESA 2005" (editors G S l Brodal, S Leonardi), Lecture Notes in Comput. Sci. 3669, Springer (2005) 95 MR2257930
14 D B A Epstein, Curves on 2-manifolds and isotopies, Acta Math. 115 (1966) 83 MR214087
15 D Gabai, Foliations and the topology of 3-manifolds, II, J. Differential Geom. 26 (1987) 461 MR910017
16 J Geelen, B Gerards, N Robertson, G Whittle, Obstructions to branch-decomposition of matroids, J. Combin. Theory Ser. B 96 (2006) 560 MR2232391
17 M Grohe, Tangles and connectivity in graphs, from: "Language and automata theory and applications" (editors A H Dediu, J Janoušek, C Martín-Vide, B Truthe), Lecture Notes in Comput. Sci. 9618, Springer (2016) 24 MR3492468
18 M Gromov, Filling Riemannian manifolds, J. Differential Geom. 18 (1983) 1 MR697984
19 M Gromov, L Guth, Generalizations of the Kolmogorov–Barzdin embedding estimates, Duke Math. J. 161 (2012) 2549 MR2988903
20 J Hass, J C Lagarias, N Pippenger, The computational complexity of knot and link problems, J. ACM 46 (1999) 185 MR1693203
21 A Hatcher, Algebraic topology, Cambridge Univ. Press (2002) MR1867354
22 C Hayashi, K Shimokawa, Thin position of a pair (3-manifold, 1-submanifold), Pacific J. Math. 197 (2001) 301 MR1815259
23 Q He, S A Taylor, Links, bridge number, and width trees, J. Math. Soc. Japan 75 (2023) 73 MR4539010
24 K Huszár, J Spreer, 3-manifold triangulations with small treewidth, from: "35th International Symposium on Computational Geometry" (editors G Barequet, Y Wang), LIPIcs. Leibniz Int. Proc. Inform. 129, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern (2019) 44 MR3968630
25 K Huszár, J Spreer, U Wagner, On the treewidth of triangulated 3-manifolds, J. Comput. Geom. 10 (2019) 70 MR4039886
26 H Jacob, M Pilipczuk, Bounding twin-width for bounded-treewidth graphs, planar graphs, and bipartite graphs, from: "Graph-theoretic concepts in computer science" (editors M A Bekos, M Kaufmann), Lecture Notes in Comput. Sci. 13453, Springer (2022) 287 MR4606560
27 F Jaeger, D L Vertigan, D J A Welsh, On the computational complexity of the Jones and Tutte polynomials, Math. Proc. Cambridge Philos. Soc. 108 (1990) 35 MR1049758
28 M Lackenby, The efficient certification of knottedness and Thurston norm, Adv. Math. 387 (2021) 107796 MR4274879
29 M Lackenby, Unknot recognition in quasi-polynomial time, lecture slides (2021)
30 M Lackenby, Algorithms in 3-manifold theory, from: "Surveys in differential geometry 2020 : surveys in 3-manifold topology and geometry" (editors I Agol, D Gabai), Surv. Differ. Geom. 25, Int. Press, Boston, MA (2022) 163 MR4479752
31 L Lyaudet, F Mazoit, S Thomassé, Partitions versus sets : a case of duality, European J. Combin. 31 (2010) 681 MR2587020
32 J A Makowsky, J P Mariño, The parametrized complexity of knot polynomials, J. Comput. System Sci. 67 (2003) 742 MR2036511
33 C Maria, Parameterized complexity of quantum knot invariants, from: "37th International Symposium on Computational Geometry" (editors K Buchin, É Colin de Verdière), LIPIcs. Leibniz Int. Proc. Inform. 189, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern (2021) 53 MR4287060
34 C Maria, J S Purcell, Treewidth, crushing and hyperbolic volume, Algebr. Geom. Topol. 19 (2019) 2625 MR4023324
35 A de Mesmay, J Purcell, S Schleimer, E Sedgwick, On the tree-width of knot diagrams, J. Comput. Geom. 10 (2019) 164 MR3957223
36 M Ozawa, Bridge position and the representativity of spatial graphs, Topology Appl. 159 (2012) 936 MR2876700
37 J Pardon, On the distortion of knots on embedded surfaces, Ann. of Math. (2) 174 (2011) 637 MR2811613
38 N Robertson, P D Seymour, Graph minors, V : Excluding a planar graph, J. Combin. Theory Ser. B 41 (1986) 92 MR854606
39 N Robertson, P D Seymour, Graph minors, X : Obstructions to tree-decomposition, J. Combin. Theory Ser. B 52 (1991) 153 MR1110468
40 N Robertson, P D Seymour, Graph minors, XI : Circuits on a surface, J. Combin. Theory Ser. B 60 (1994) 72 MR1256585
41 N Robertson, P D Seymour, Graph minors, XX : Wagner’s conjecture, J. Combin. Theory Ser. B 92 (2004) 325 MR2099147
42 H Sachs, On a spatial analogue of Kuratowski’s theorem on planar graphs—an open problem, from: "Graph theory" (editors M Borowiecki, J W Kennedy, M M Sysło), Lecture Notes in Math. 1018, Springer (1983) 230 MR730653
43 M Scharlemann, Thin position in the theory of classical knots, from: "Handbook of knot theory" (editors W Menasco, M Thistlethwaite), Elsevier B. V., Amsterdam (2005) 429 MR2179267
44 M Scharlemann, J Schultens, T Saito, Lecture notes on generalized Heegaard splittings, World Scientific, Hackensack, NJ (2016) MR3585907
45 M Scharlemann, A Thompson, Thin position for 3-manifolds, from: "Geometric Topology: Joint US-Israel Workshop on Geometric Topology", Contemporary Mathematics 164, Amer. Math. Soc. (1994) 231
46 J Schultens, Introduction to 3-manifolds, 151, Amer. Math. Soc. (2014) MR3203728
47 J Schultens, The bridge number of a knot, from: "Encyclopedia of Knot Theory" (editors C Adams, E Flapan, A Henrich, L H Kauffman, L Ludwig, S Nelson), Chapman and Hall/CRC (2021) 229
48 P D Seymour, R Thomas, Call routing and the ratcatcher, Combinatorica 14 (1994) 217 MR1289074
49 S Taylor, M Tomova, Additive invariants for knots, links and graphs in 3-manifolds, Geom. Topol. 22 (2018) 3235 MR3858764
50 A Turing, Solvable and unsolvable problems, magazine article (1954)