Algebraic & Geometric Topology Volume 19, issue 7 (2019)

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the Journal

Editorial Board

Editorial Interests

Subscriptions

Submission Guidelines

Submission Page

Policies for Authors

Ethics Statement

ISSN (electronic): 1472-2739

ISSN (print): 1472-2747

Author Index

To Appear

Other MSP Journals

Ropelength, crossing number and finite-type invariants of links

Rafal Komendarczyk and Andreas Michaelides

Algebraic & Geometric Topology 19 (2019) 3335–3357

DOI: 10.2140/agt.2019.19.3335

Bibliography

1	M Abramowitz, I A Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, 55, US Government Printing Office (1964) MR0167642
2	D Bar-Natan, Perturbative aspects of the Chern–Simons topological quantum field theory, PhD thesis, Princeton University (1991) MR2686517
3	D Bar-Natan, Polynomial invariants are polynomial, Math. Res. Lett. 2 (1995) 239 MR1338783
4	R Bott, C Taubes, On the self-linking of knots, J. Math. Phys. 35 (1994) 5247 MR1295465
5	G Buck, J Simon, Thickness and crossing number of knots, Topology Appl. 91 (1999) 245 MR1666650
6	G R Buck, J K Simon, Total curvature and packing of knots, Topology Appl. 154 (2007) 192 MR2271783
7	R Buniy, J Cantarella, T Kephart, E Rawdon, Tight knot spectrum in QCD, Phys. Rev. D 89 (2014)
8	J. Cantarella, Introduction to geometric knot theory, I: Knot invariants defined by minimizing curve invariants, talk slides (2009)
9	J. Cantarella, Introduction to geometric knot theory, II: Ropelength and tight knots, talk slides (2009)
10	J Cantarella, R B Kusner, J M Sullivan, On the minimum ropelength of knots and links, Invent. Math. 150 (2002) 257 MR1933586
11	S Chmutov, S Duzhin, J Mostovoy, Introduction to Vassiliev knot invariants, Cambridge Univ. Press (2012) MR2962302
12	S Chmutov, M C Khoury, A Rossi, Polyak-viro formulas for coefficients of the Conway polynomial, J. Knot Theory Ramifications 18 (2009) 773 MR2542695
13	Y Diao, The lower bounds of the lengths of thick knots, J. Knot Theory Ramifications 12 (2003) 1 MR1953620
14	Y Diao, C Ernst, Total curvature, ropelength and crossing number of thick knots, Math. Proc. Cambridge Philos. Soc. 143 (2007) 41 MR2340974
15	Y Diao, C Ernst, E J Janse Van Rensburg, Upper bounds on linking numbers of thick links, J. Knot Theory Ramifications 11 (2002) 199 MR1895370
16	Y Diao, C Ernst, E J Janse van Rensburg, Thicknesses of knots, Math. Proc. Cambridge Philos. Soc. 126 (1999) 293 MR1670241
17	C Ernst, A Por, Average crossing number, total curvature and ropelength of thick knots, J. Knot Theory Ramifications 21 (2012) MR2887660
18	H Federer, Geometric measure theory, 153, Springer (1969) MR0257325
19	M H Freedman, Z X He, Divergence-free fields: energy and asymptotic crossing number, Ann. of Math. 134 (1991) 189 MR1114611
20	M Freedman, V Krushkal, Geometric complexity of embeddings in ℝ^d, Geom. Funct. Anal. 24 (2014) 1406 MR3261630
21	P M González Manchón, Homogeneous links and the Seifert matrix, Pacific J. Math. 255 (2012) 373 MR2928557
22	M Goussarov, M Polyak, O Viro, Finite-type invariants of classical and virtual knots, Topology 39 (2000) 1045 MR1763963
23	A Gray, Tubes, 221, Birkhäuser (2004) MR2024928
24	M Gromov, Filling Riemannian manifolds, J. Differential Geom. 18 (1983) 1 MR697984
25	M Gromov, L Guth, Generalizations of the Kolmogorov–Barzdin embedding estimates, Duke Math. J. 161 (2012) 2549 MR2988903
26	A Kawauchi, A survey of knot theory, Birkhäuser (1996) MR1417494
27	R Komendarczyk, A Michaelides, Tree invariants and Milnor linking numbers with indeterminacy, preprint (2016) arXiv:1602.06449
28	Y Kotorii, The Milnor μ invariants and nanophrases, J. Knot Theory Ramifications 22 (2013) MR3037291
29	O Kravchenko, M Polyak, Diassociative algebras and Milnor’s invariants for tangles, Lett. Math. Phys. 95 (2011) 297 MR2775128
30	E H Lieb, M Loss, Analysis, 14, Amer. Math. Soc. (2001) MR1817225
31	X S Lin, Z Wang, Integral geometry of plane curves and knot invariants, J. Differential Geom. 44 (1996) 74 MR1420350
32	R A Litherland, J Simon, O Durumeric, E Rawdon, Thickness of knots, Topology Appl. 91 (1999) 233 MR1666649
33	F Maggioni, R L Ricca, On the groundstate energy of tight knots, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465 (2009) 2761 MR2524831
34	A V Michaelides, Lower bounds for ropelength of links via higher linking numbers and other finite type invariants, PhD thesis, Tulane University School of Science and Engineering (2015) MR3579391
35	J Milnor, Link groups, Ann. of Math. 59 (1954) 177 MR0071020
36	J Milnor, Isotopy of links: algebraic geometry and topology, from: "A symposium in honor of S Lefschetz" (editors R H Fox, D C Spencer, A W Tucker), Princeton Univ. Press (1957) 280 MR0092150
37	M Polyak, On the algebra of arrow diagrams, Lett. Math. Phys. 51 (2000) 275 MR1778064
38	M Polyak, O Viro, Gauss diagram formulas for Vassiliev invariants, Int. Math. Res. Not. 1994 (1994) 445 MR1316972
39	M Polyak, O Viro, On the Casson knot invariant, J. Knot Theory Ramifications 10 (2001) 711 MR1839698
40	E J Rawdon, Approximating the thickness of a knot, from: "Ideal knots" (editors A Stasiak, V Katritch, L H Kauffman), Ser. Knots Everything 19, World Sci. (1998) 143 MR1702029
41	R L Ricca, F Maggioni, On the groundstate energy spectrum of magnetic knots and links, J. Phys. A 47 (2014) MR3200528
42	R L Ricca, H K Moffatt, The helicity of a knotted vortex filament, from: "Topological aspects of the dynamics of fluids and plasmas" (editors H K Moffatt, G M Zaslavsky, P Comte, M Tabor), NATO Adv. Sci. Inst. Ser. E Appl. Sci. 218, Kluwer (1992) 225 MR1232233
43	I Volić, A survey of Bott–Taubes integration, J. Knot Theory Ramifications 16 (2007) 1 MR2300426
44	S Willerton, On the first two Vassiliev invariants, Experiment. Math. 11 (2002) 289 MR1959269