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The unknotting number and classical invariants, I

Maciej Borodzik and Stefan Friedl

Algebraic & Geometric Topology 15 (2015) 85–135
Bibliography
1 J A Baldwin, W D Gillam, Computations of Heegaard–Floer knot homology, J. Knot Theory Ramifications 21 (2012) 1250075, 65 MR2925428
2 R C Blanchfield, Intersection theory of manifolds with operators with applications to knot theory, Ann. of Math. 65 (1957) 340 MR0085512
3 M Boileau, C Weber, Le problème de J Milnor sur le nombre gordien des nœuds algébriques, Enseign. Math. 30 (1984) 173 MR767901
4 M Borodzik, S Friedl, Knotorious world wide web page
5 M Borodzik, S Friedl, On the algebraic unknotting number, Trans. Lon. Math. Soc. 1 (2014) 57
6 M Borodzik, S Friedl, The unknotting number and classical invariants, II, Glasg. Math. J. 56 (2014) 657 MR3250270
7 R D Brandt, W B R Lickorish, K C Millett, A polynomial invariant for unoriented knots and links, Invent. Math. 84 (1986) 563 MR837528
8 J W S Cassels, Rational quadratic forms, 13, Academic Press (1978) MR522835
9 J C Cha, C Livingston, KnotInfo : Table of knot invariants
10 T D Cochran, W B R Lickorish, Unknotting information from 4–manifolds, Trans. Amer. Math. Soc. 297 (1986) 125 MR849471
11 J H Conway, N J A Sloane, Sphere packings, lattices and groups, 290, Springer (1999) MR1662447
12 D Cooper, W B R Lickorish, Mutations of links in genus-2 handlebodies, Proc. Amer. Math. Soc. 127 (1999) 309 MR1605940
13 M E Fogel, The algebraic unknotting number, PhD thesis, UC Berkeley (1993)
14 M E Fogel, Knots with algebraic unknotting number one, Pacific J. Math. 163 (1994) 277 MR1262298
15 M H Freedman, F Quinn, Topology of 4–manifolds, 39, Princeton Univ. Press (1990) MR1201584
16 S Friedl, I Hambleton, P Melvin, P Teichner, Nonsmoothable four-manifolds with infinite cyclic fundamental group, Int. Math. Res. Not. 2007 (2007) MR2344268
17 C M Gordon, Some aspects of classical knot theory, from: "Knot theory" (editor J C Hausmann), Lecture Notes in Math. 685, Springer (1978) 1 MR521730
18 C M Gordon, J Luecke, Knots with unknotting number 1 and essential Conway spheres, Algebr. Geom. Topol. 6 (2006) 2051 MR2263059
19 J E Greene, Donaldson’s theorem, Heegaard–Floer homology, and knots with unknotting number one, Adv. Math. 255 (2014) 672 MR3167496
20 J Hillman, Algebraic invariants of links, 52, World Scientific (2012) MR2931688
21 C F Ho, A polynomial invariant for knots and links : Preliminary report (1985)
22 D Husemoller, J Milnor, Symmetric bilinear forms, Springer (1973) MR0506372
23 S Jabuka, The rational Witt class and the unknotting number of a knot, Preprint (2009)
24 T Kanenobu, H Murakami, Two-bridge knots with unknotting number one, Proc. Amer. Math. Soc. 98 (1986) 499 MR857949
25 A Kawauchi, Three dualities on the integral homology of infinite cyclic coverings of manifolds, Osaka J. Math. 23 (1986) 633 MR866269
26 A Kawauchi, A survey of knot theory, Birkhäuser (1996) MR1417494
27 C Kearton, Blanchfield duality and simple knots, Trans. Amer. Math. Soc. 202 (1975) 141 MR0358796
28 C Kearton, Mutation of knots, Proc. Amer. Math. Soc. 105 (1989) 206 MR929430
29 C Kearton, S–equivalence of knots, J. Knot Theory Ramifications 13 (2004) 709 MR2088740
30 C Kearton, S M J Wilson, Knot modules and the Nakanishi index, Proc. Amer. Math. Soc. 131 (2003) 655 MR1933359
31 S Kinoshita, On Wendt’s theorem of knots, Osaka Math. J. 9 (1957) 61 MR0090054
32 S Kinoshita, On Wendt’s theorem of knots, II, Osaka Math. J. 10 (1958) 259 MR0124889
33 R Kirby, Problems in low-dimensional topology, from: "Geometric topology" (editor W H Kazez), AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc. (1997) 35 MR1470751
34 P Kirk, C Livingston, Concordance and mutation, Geom. Topol. 5 (2001) 831 MR1871406
35 K H Ko, A Seifert-matrix interpretation of Cappell and Shaneson’s approach to link cobordisms, Math. Proc. Cambridge Philos. Soc. 106 (1989) 531 MR1010376
36 T Kobayashi, Minimal genus Seifert surfaces for unknotting number 1 knots, Kobe J. Math. 6 (1989) 53 MR1023526
37 P B Kronheimer, T S Mrowka, Gauge theory for embedded surfaces, I, Topology 32 (1993) 773 MR1241873
38 C Leidy, Higher-order linking forms for knots, Comment. Math. Helv. 81 (2006) 755 MR2271220
39 J Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969) 229 MR0246314
40 J Levine, An algebraic classification of some knots of codimension two, Comment. Math. Helv. 45 (1970) 185 MR0266226
41 J Levine, Knot modules, I, Trans. Amer. Math. Soc. 229 (1977) 1 MR0461518
42 W B R Lickorish, The unknotting number of a classical knot, from: "Combinatorial methods in topology and algebraic geometry" (editors J R Harper, R Mandelbaum), Contemp. Math. 44, Amer. Math. Soc. (1985) 117 MR813107
43 W B R Lickorish, An introduction to knot theory, 175, Springer (1997) MR1472978
44 C Livingston, Knot 4–genus and the rank of classes in W((t)), Pacific J. Math. 252 (2011) 113 MR2862144
45 J Ma, R Qiu, A lower bound on unknotting number, Chinese Ann. Math. Ser. B 27 (2006) 437 MR2250868
46 Y Miyazawa, The Jones polynomial of an unknotting number one knot, Topol. Appl. 83 (1998) 161 MR1606374
47 J M Montesinos, Seifert manifolds that are ramified two-sheeted cyclic coverings, Bol. Soc. Mat. Mexicana 18 (1973) 1 MR0341467
48 H Murakami, Algebraic unknotting operation, from: "Proceedings of the Second Soviet–Japan Joint Symposium of Topology", Questions Answers Gen. Topol. 8 (1990) 283 MR1043226
49 K Murasugi, On a certain numerical invariant of link types, Trans. Amer. Math. Soc. 117 (1965) 387 MR0171275
50 Y Nakanishi, A note on unknotting number, Math. Sem. Notes Kobe Univ. 9 (1981) 99 MR634000
51 B Owens, Unknotting information from Heegaard Floer homology, Adv. Math. 217 (2008) 2353 MR2388097
52 B Owens, S Strle, Definite manifolds bounded by rational homology three spheres, from: "Geometry and topology of manifolds" (editors H U Boden, I Hambleton, A J Nicas, B D Park), Fields Inst. Commun. 47, Amer. Math. Soc. (2005) 243 MR2189936
53 P Ozsváth, Z Szabó, Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary, Adv. Math. 173 (2003) 179 MR1957829
54 P Ozsváth, Z Szabó, Knot Floer homology and the four-ball genus, Geom. Topol. 7 (2003) 615 MR2026543
55 P Ozsváth, Z Szabó, Knots with unknotting number one and Heegaard Floer homology, Topology 44 (2005) 705 MR2136532
56 A Ranicki, Exact sequences in the algebraic theory of surgery, 26, Princeton Univ. Press (1981) MR620795
57 A Ranicki, Blanchfield and Seifert algebra in high-dimensional knot theory, Mosc. Math. J. 3 (2003) 1333 MR2058802
58 J A Rasmussen, Floer homology and knot complements, PhD thesis, Harvard University (2003)
59 J Rasmussen, Khovanov homology and the slice genus, Invent. Math. 182 (2010) 419 MR2729272
60 O Saeki, On algebraic unknotting numbers of knots, Tokyo J. Math. 22 (1999) 425 MR1727885
61 S Sarkar, Grid diagrams and the Ozsváth–Szabó tau-invariant, Math. Res. Lett. 18 (2011) 1239 MR2915478
62 M G Scharlemann, Unknotting number one knots are prime, Invent. Math. 82 (1985) 37 MR808108
63 M G Scharlemann, A Thompson, Link genus and the Conway moves, Comment. Math. Helv. 64 (1989) 527 MR1022995
64 A Stoimenow, Polynomial values, the linking form and unknotting numbers, Math. Res. Lett. 11 (2004) 755 MR2106240
65 L R Taylor, On the genera of knots, from: "Topology of low-dimensional manifolds" (editor R A Fenn), Lecture Notes in Math. 722, Springer (1979) 144 MR547461
66 L R Taylor, Relative Rochlin invariants, Topol. Appl. 18 (1984) 259 MR769295
67 P Traczyk, A criterion for signed unknotting number, from: "Low-dimensional topology" (editor H Nencka), Contemp. Math. 233, Amer. Math. Soc. (1999) 215 MR1701685
68 A G Tristram, Some cobordism invariants for links, Proc. Cambridge Philos. Soc. 66 (1969) 251 MR0248854
69 H F Trotter, On S–equivalence of Seifert matrices, Invent. Math. 20 (1973) 173 MR0645546
70 B L van der Waerden, Die Reduktionstheorie der positiven quadratischen Formen, Acta Math. 96 (1956) 265 MR0082513
71 H Wendt, Die gordische Auflösung von Knoten, Math. Z. 42 (1937) 680 MR1545700