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Milnor's concordance invariants for knots on surfaces

Micah Chrisman

Algebraic & Geometric Topology 22 (2022) 2293–2353
Bibliography
1 B Audoux, P Bellingeri, J B Meilhan, E Wagner, Homotopy classification of ribbon tubes and welded string links, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 17 (2017) 713 MR3700382
2 D Bar-Natan, Vassiliev homotopy string link invariants, J. Knot Theory Ramifications 4 (1995) 13 MR1321289
3 D Bar-Natan, Crossing the crossings, lecture (2015)
4 V G Bardakov, P Bellingeri, Groups of virtual and welded links, J. Knot Theory Ramifications 23 (2014) MR3200494
5 H U Boden, M Chrisman, Virtual concordance and the generalized Alexander polynomial, J. Knot Theory Ramifications 30 (2021) MR4284604
6 H U Boden, M Chrisman, R Gaudreau, Virtual knot cobordism and bounding the slice genus, Exp. Math. 28 (2019) 475 MR4032961
7 H U Boden, M Chrisman, R Gaudreau, Signature and concordance of virtual knots, Indiana Univ. Math. J. 69 (2020) 2395 MR4195608
8 H U Boden, E Dies, A I Gaudreau, A Gerlings, E Harper, A J Nicas, Alexander invariants for virtual knots, J. Knot Theory Ramifications 24 (2015) MR3342135
9 H U Boden, R Gaudreau, E Harper, A J Nicas, L White, Virtual knot groups and almost classical knots, Fund. Math. 238 (2017) 101 MR3640614
10 H U Boden, M Nagel, Concordance group of virtual knots, Proc. Amer. Math. Soc. 145 (2017) 5451 MR3717970
11 J S Carter, S Kamada, M Saito, Stable equivalence of knots on surfaces and virtual knot cobordisms, J. Knot Theory Ramifications 11 (2002) 311 MR1905687
12 K T Chen, R H Fox, R C Lyndon, Free differential calculus, IV : The quotient groups of the lower central series, Ann. of Math. (2) 68 (1958) 81 MR102539
13 Z Cheng, H Gao, A polynomial invariant of virtual links, J. Knot Theory Ramifications 22 (2013) MR3149308
14 M Chrisman, Band-passes and long virtual knot concordance, J. Knot Theory Ramifications 26 (2017) MR3692566
15 D Cimasoni, V Turaev, A generalization of several classical invariants of links, Osaka J. Math. 44 (2007) 531 MR2360939
16 T D Cochran, Derivatives of links : Milnor’s concordance invariants and Massey’s products, 427, Amer. Math. Soc. (1990) MR1042041
17 T D Cochran, K E Orr, Not all links are concordant to boundary links, Ann. of Math. (2) 138 (1993) 519 MR1247992
18 T D Cochran, K E Orr, P Teichner, Knot concordance, Whitney towers and L2–signatures, Ann. of Math. (2) 157 (2003) 433 MR1973052
19 A S Crans, A Henrich, S Nelson, Polynomial knot and link invariants from the virtual biquandle, J. Knot Theory Ramifications 22 (2013) MR3055555
20 H A Dye, A Kaestner, L H Kauffman, Khovanov homology, Lee homology and a Rasmussen invariant for virtual knots, J. Knot Theory Ramifications 26 (2017) MR3627701
21 H A Dye, L H Kauffman, Virtual homotopy, J. Knot Theory Ramifications 19 (2010) 935 MR2673693
22 R A Fenn, Techniques of geometric topology, 57, Cambridge Univ. Press (1983) MR787801
23 T G Group, GAP: groups, algorithms, and programming, software (2019)
24 J Green, A table of virtual knots, electronic resource (2004)
25 N Habegger, X S Lin, The classification of links up to link-homotopy, J. Amer. Math. Soc. 3 (1990) 389 MR1026062
26 N Habegger, X S Lin, On link concordance and Milnor’s μ invariants, Bull. Lond. Math. Soc. 30 (1998) 419 MR1620841
27 N Habegger, G Masbaum, The Kontsevich integral and Milnor’s invariants, Topology 39 (2000) 1253 MR1783857
28 M Hall Jr., The theory of groups, Macmillan (1959) MR0103215
29 A Henrich, A sequence of degree one Vassiliev invariants for virtual knots, J. Knot Theory Ramifications 19 (2010) 461 MR2646641
30 J A Hillman, Alexander ideals and Chen groups, Bull. Lond. Math. Soc. 10 (1978) 105 MR478132
31 F Jaeger, L H Kauffman, H Saleur, The Conway polynomial in 3 and in thickened surfaces : a new determinant formulation, J. Combin. Theory Ser. B 61 (1994) 237 MR1280610
32 H Karimi, The Khovanov homology of alternating virtual links, Michigan Math. J. 70 (2021) 749 MR4332676
33 L H Kauffman, Virtual knot theory, European J. Combin. 20 (1999) 663 MR1721925
34 L H Kauffman, A self-linking invariant of virtual knots, Fund. Math. 184 (2004) 135 MR2128048
35 L H Kauffman, An affine index polynomial invariant of virtual knots, J. Knot Theory Ramifications 22 (2013) MR3055558
36 L H Kauffman, Virtual knot cobordism, from: "New ideas in low dimensional topology" (editors L H Kauffman, V O Manturov), Ser. Knots Everything 56, World Sci. (2015) 335 MR3381329
37 L H Kauffman, D Radford, Bi-oriented quantum algebras, and a generalized Alexander polynomial for virtual links, from: "Diagrammatic morphisms and applications" (editors D E Radford, F J O Souza, D N Yetter), Contemp. Math. 318, Amer. Math. Soc. (2003) 113 MR1973514
38 G Kuperberg, What is a virtual link?, Algebr. Geom. Topol. 3 (2003) 587 MR1997331
39 X S Lin, Power series expansions and invariants of links, from: "Geometric topology" (editor W H Kazez), AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc. (1997) 184 MR1470727
40 W Magnus, A Karrass, D Solitar, Combinatorial group theory: presentations of groups in terms of generators and relations, Interscience (1966) MR0207802
41 V O Manturov, On invariants of virtual links, Acta Appl. Math. 72 (2002) 295 MR1916950
42 V O Manturov, Compact and long virtual knots, Tr. Mosk. Mat. Obs. 69 (2008) 5 MR2549444
43 B Mellor, Alexander and writhe polynomials for virtual knots, J. Knot Theory Ramifications 25 (2016) MR3530309
44 J Milnor, Isotopy of links, from: "Algebraic geometry and topology : a symposium in honor of S Lefschetz" (editors R H Fox, D C Spencer, A W Tucker), Princeton Math. Ser. 12, Princeton Univ. Press (1957) 280 MR0092150
45 K Y Ng, Groups of ribbon knots, Topology 37 (1998) 441 MR1489214
46 W Nickel, Computing nilpotent quotients of finitely presented groups, from: "Geometric and computational perspectives on infinite groups" (editors G Baumslag, D Epstein, R Gilman, H Short, C Sims), DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 25, Amer. Math. Soc. (1996) 175 MR1364184
47 K E Orr, Homotopy invariants of links, Invent. Math. 95 (1989) 379 MR974908
48 M Polyak, Minimal generating sets of Reidemeister moves, Quantum Topol. 1 (2010) 399 MR2733246
49 D Roseman, Reidemeister-type moves for surfaces in four-dimensional space, from: "Knot theory" (editors V F R Jones, J Kania-Bartoszyńska, J H Przytycki, P Traczyk, V G Turaev), Banach Center Publ. 42, Polish Acad. Sci. Inst. Math. (1998) 347 MR1634466
50 W Rushworth, Computations of the slice genus of virtual knots, Topology Appl. 253 (2019) 57 MR3892369
51 S Satoh, Virtual knot presentation of ribbon torus-knots, J. Knot Theory Ramifications 9 (2000) 531 MR1758871
52 J Sawollek, On Alexander–Conway polynomials for virtual knots and links, preprint (1999) arXiv:math/9912173
53 D S Silver, S G Williams, Virtual knot groups, from: "Knots in Hellas ’98" (editors C M Gordon, V F R Jones, L H Kauffman, S Lambropoulou, J H Przytycki), Ser. Knots Everything 24, World Sci. (2000) 440 MR1865722
54 D S Silver, S G Williams, Alexander groups and virtual links, J. Knot Theory Ramifications 10 (2001) 151 MR1822148
55 D S Silver, S G Williams, Polynomial invariants of virtual links, J. Knot Theory Ramifications 12 (2003) 987 MR2017967
56 D S Silver, S G Williams, Crowell’s derived group and twisted polynomials, J. Knot Theory Ramifications 15 (2006) 1079 MR2275098
57 J Stallings, Homology and central series of groups, J. Algebra 2 (1965) 170 MR175956
58 L Traldi, Milnor’s invariants and the completions of link modules, Trans. Amer. Math. Soc. 284 (1984) 401 MR742432
59 V Turaev, Cobordism of knots on surfaces, J. Topol. 1 (2008) 285 MR2399131