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Noncommutative knot theory. (English) Zbl 1063.57011

The aim of this paper is to introduce new invariants for knots, which are higher order analogs of the classical Alexander module and Alexander polynomial. Given a knot \(K\) in \(S^3\), let \(G\) be its knot group, and for \(n\geq 0\) let \(G^{(n+1)}\) be the \((n+1)\)-st term in the derived series of \(G\). The \(n\)-th integral Alexander module \(\mathcal A_n^{\mathbb Z}(K)\) is defined as \(G^{(n+1)}/G^{(n+2)}\), viewed as a right module over the ring \(\mathbb Z[G/G^{(n+1)}]\). For \(n>0\), this ring is non-commutative, non-Noetherian and not an UFD in general, which a priori makes handling these modules difficult. In spite of these problems, the author shows that these modules lead to effective invariants with applications to estimating knot genus, detecting fibered, prime, and alternating knots, and producing new obstructions to the existence of symplectic structures on certain four manifolds.
To deal with modules over \(\mathbb Z[G/G^{(n+1)}]\), the author uses the theory of poly-(torsion-free abelian) groups. Via a localized version \(\mathcal A_n(K)\) of \(\mathcal A_n^{\mathbb Z}(K)\), he introduces the \(n\)-th order Alexander polynomial. While the indeterminacy in this polynomial makes it difficult to handle, its degree is a well defined numerical knot invariant \(\delta_n(K)\). This generalizes the classical Alexander module and Alexander polynomial, which is obtained for \(n=0\). The fundamental result for most of the applications discussed in the paper is then that for a non-trivial knot \(K\) and each \(n>0\), one has \(\delta_0(K)\leq\delta_1(K)+1\leq\dots\leq\delta_n(K)+1\leq 2\cdot\text{genus}(K)\). If \(K\) is fibered or alternating or if the \(0\)-surgery \(M_K\) at \(K\) has the property that \(S^1\times M_K\) admits a symplectic structure, then all these inequalities have to be equalities.
An important part of the paper is devoted to a construction for modifications of a knot, called genetic infection, whose influence on the higher order Alexander module can be well controlled. For a knot \(K\) with non-trivial classical Alexander polynomial, this leads to families of modifications, whose effect on \(\mathcal A_n^{\mathbb Z}\) and on \(\mathcal A_n\) can be controlled very well. In particular, one obtains modifications for which some of the above inequalities become proper.
Finally, the author also studies higher order bordism invariants generalizing the Arf invariant, higher order von Neumann signatures of knots, and higher order generalizations of the Blanchfield linking forms.
Reviewer: Andreas Cap (Wien)

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
20F14 Derived series, central series, and generalizations for groups
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References:

[1] M F Atiyah, Elliptic operators, discrete groups and von Neumann algebras, Soc. Math. France (1976) · Zbl 0323.58015
[2] M Boileau, S Wang, Non-zero degree maps and surface bundles over \(S^1\), J. Differential Geom. 43 (1996) 789 · Zbl 0868.57029
[3] K S Brown, Cohomology of groups, Graduate Texts in Mathematics 87, Springer (1982) · Zbl 0584.20036
[4] J C Cha, Fibred knots and twisted Alexander invariants, Trans. Amer. Math. Soc. 355 (2003) 4187 · Zbl 1028.57004 · doi:10.1090/S0002-9947-03-03348-8
[5] T D Cochran, K E Orr, Homology boundary links and Blanchfield forms: concordance classification and new tangle-theoretic constructions, Topology 33 (1994) 397 · Zbl 0828.57016 · doi:10.1016/0040-9383(94)90020-5
[6] T D Cochran, K E Orr, P Teichner, Knot concordance, Whitney towers and \(L^2\)-signatures, Ann. of Math. \((2)\) 157 (2003) 433 · Zbl 1044.57001 · doi:10.4007/annals.2003.157.433
[7] T D Cochran, K E Orr, P Teichner, Structure in the classical knot concordance group, Comment. Math. Helv. 79 (2004) 105 · Zbl 1061.57008 · doi:10.1007/s00014-003-0793-6
[8] T D Cochran, P Teichner, Knot concordance and von Neumann \(\rho\)-invariants, Duke Math. J. 137 (2007) 337 · Zbl 1186.57006 · doi:10.1215/S0012-7094-07-13723-2
[9] P M Cohn, Skew fields, Encyclopedia of Mathematics and its Applications 57, Cambridge University Press (1995) · Zbl 0840.16001
[10] P M Cohn, Free rings and their relations, London Mathematical Society Monographs 19, Academic Press (1985) · Zbl 0659.16001
[11] J Cheeger, M Gromov, Bounds on the von Neumann dimension of \(L^2\)-cohomology and the Gauss-Bonnet theorem for open manifolds, J. Differential Geom. 21 (1985) 1 · Zbl 0614.53034
[12] J Conant, P Teichner, Grope cobordism of classical knots, Topology 43 (2004) 119 · Zbl 1041.57003 · doi:10.1016/S0040-9383(03)00031-4
[13] J Dodziuk, P Linnell, V Mathai, T Schick, S Yates, Approximating \(L^2\)-invariants and the Atiyah conjecture, Comm. Pure Appl. Math. 56 (2003) 839 · Zbl 1036.58017 · doi:10.1002/cpa.10076
[14] J Duval, Forme de Blanchfield et cobordisme d’entrelacs bords, Comment. Math. Helv. 61 (1986) 617 · Zbl 0632.57014 · doi:10.1007/BF02621935
[15] S Garoufalidis, L Rozansky, The loop expansion of the Kontsevich integral, the null-move and \(S\)-equivalence, Topology 43 (2004) 1183 · Zbl 1052.57011 · doi:10.1016/j.top.2004.01.002
[16] S Garoufalidis, J Levine, Homology surgery and invariants of 3-manifolds, Geom. Topol. 5 (2001) 551 · Zbl 1009.57022 · doi:10.2140/gt.2001.5.551
[17] C M Gordon, Some aspects of classical knot theory, Lecture Notes in Math. 685, Springer (1978) 1 · Zbl 0386.57002
[18] K Habiro, Claspers and finite type invariants of links, Geom. Topol. 4 (2000) 1 · Zbl 0941.57015 · doi:10.2140/gt.2000.4.1
[19] S L Harvey, Higher-order polynomial invariants of 3-manifolds giving lower bounds for the Thurston norm, Topology 44 (2005) 895 · Zbl 1080.57019 · doi:10.1016/j.top.2005.03.001
[20] J Harer, How to construct all fibered knots and links, Topology 21 (1982) 263 · Zbl 0504.57002 · doi:10.1016/0040-9383(82)90009-X
[21] J Hempel, Intersection calculus on surfaces with applications to 3-manifolds, Mem. Amer. Math. Soc. 43 (1983) · Zbl 0518.57008
[22] G Higman, The units of group-rings, Proc. London Math. Soc. \((2)\) 46 (1940) 231 · Zbl 0025.24302 · doi:10.1112/plms/s2-46.1.231
[23] J A Hillman, Alexander ideals of links, Lecture Notes in Mathematics 895, Springer (1981) · Zbl 0491.57001 · doi:10.1007/BFb0091682
[24] N Higson, G Kasparov, preprint (1998)
[25] P J Hilton, U Stammbach, A course in homological algebra, Graduate Texts in Mathematics 4, Springer (1971) · Zbl 0238.18006
[26] J Howie, H R Schneebeli, Homological and topological properties of locally indicable groups, Manuscripta Math. 44 (1983) 71 · Zbl 0533.20022 · doi:10.1007/BF01166075
[27] N Jacobson, The Theory of Rings, American Mathematical Society Mathematical Surveys I, American Mathematical Society (1943) · Zbl 0060.07302
[28] A Kawauchi, Almost identical imitations of \((3,1)\)-dimensional manifold pairs, Osaka J. Math. 26 (1989) 743 · Zbl 0701.57015
[29] R Kirby, A calculus for framed links in \(S^3\), Invent. Math. 45 (1978) 35 · Zbl 0377.55001 · doi:10.1007/BF01406222
[30] P Kirk, C Livingston, Twisted Alexander invariants, Reidemeister torsion, and Casson-Gordon invariants, Topology 38 (1999) 635 · Zbl 0928.57005 · doi:10.1016/S0040-9383(98)00039-1
[31] P B Kronheimer, Embedded surfaces and gauge theory in three and four dimensions, Int. Press, Boston (1998) 243 · Zbl 0965.57030
[32] P B Kronheimer, Minimal genus in \(S^1{\times}M^3\), Invent. Math. 135 (1999) 45 · Zbl 0917.57017 · doi:10.1007/s002220050279
[33] J Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969) 229 · Zbl 0176.22101 · doi:10.1007/BF02564525
[34] J Levine, Knot modules I, Trans. Amer. Math. Soc. 229 (1977) 1 · Zbl 0653.57012 · doi:10.2307/1998498
[35] J Lewin, A note on zero divisors in group-rings, Proc. Amer. Math. Soc. 31 (1972) 357 · Zbl 0242.16006 · doi:10.2307/2037530
[36] V Mathai, Spectral flow, eta invariants, and von Neumann algebras, J. Funct. Anal. 109 (1992) 442 · Zbl 0783.57015 · doi:10.1016/0022-1236(92)90022-B
[37] J R Munkres, Elements of algebraic topology, Addison-Wesley Publishing Company (1984) · Zbl 0673.55001
[38] J R Munkres, Topology: a first course, Prentice-Hall (1975) · Zbl 0306.54001
[39] D S Passman, The algebraic structure of group rings, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons] (1977) · Zbl 0368.16003
[40] S Naik, T Stanford, A move on diagrams that generates \(S\)-equivalence of knots, J. Knot Theory Ramifications 12 (2003) 717 · Zbl 1051.57010 · doi:10.1142/S0218216503002639
[41] A Ranicki, Exact sequences in the algebraic theory of surgery, Mathematical Notes 26, Princeton University Press (1981) · Zbl 0471.57012
[42] S Roushon, Topology of 3-manifolds and a class of groups · Zbl 1134.57302
[43] J R Stallings, Constructions of fibred knots and links, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc. (1978) 55 · Zbl 0394.57007
[44] J Stallings, Homology and central series of groups, J. Algebra 2 (1965) 170 · Zbl 0135.05201 · doi:10.1016/0021-8693(65)90017-7
[45] B Stenström, Rings of quotients, Grundlehren der Mathematischen Wissenschaften 217, Springer (1975) · Zbl 0296.16001
[46] A Stoimenow, Vassiliev invariants on fibered and mutually obverse knots, J. Knot Theory Ramifications 8 (1999) 511 · Zbl 0937.57008 · doi:10.1142/S0218216599000353
[47] R Strebel, Homological methods applied to the derived series of groups, Comment. Math. Helv. 49 (1974) 302 · Zbl 0288.20066
[48] G W Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics 61, Springer (1978) · Zbl 0406.55001
[49] C T C Wall, Surgery on compact manifolds, London Mathematical Society Monographs 1, Academic Press (1970) · Zbl 0219.57024
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