Surgery presentations of coloured knots and of their covering links

Kricker, Andrew¹ ; Moskovich, Daniel²

¹School of Physical & Mathematical Sciences, Nanyang Technological University, SPMS-04-01, 21 Nanyang Link, Singapore 637371
²Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan

Algebraic and Geometric Topology, Tome 9 (2009) no. 3, pp. 1341-1398

Résumé

We consider knots equipped with a representation of their knot groups onto a dihedral group D2n (where n is odd). To each such knot there corresponds a closed 3–manifold, the (irregular) dihedral branched covering space, with the branching set over the knot forming a link in it. We report a variety of results relating to the problem of passing from the initial data of a D2n–coloured knot to a surgery presentation of the corresponding branched covering space and covering link. In particular, we describe effective algorithms for constructing such presentations. A by-product of these investigations is a proof of the conjecture that two D2n–coloured knots are related by a sequence of surgeries along ± 1–framed unknots in the kernel of the representation if and only if they have the same coloured untying invariant (a ℤn–valued algebraic invariant of D2n–coloured knots).

Détail

DOI : 10.2140/agt.2009.9.1341

Keywords: dihedral covering, covering space, covering linkage, Fox $n$–colouring, surgery presentation

Bibliographie

[1] J S Birman, A note on the construction of simply-connected $3$–manifolds as branched covering spaces of $S^{3}$, Proc. Amer. Math. Soc. 55 (1976) 440

[2] G Burde, H Zieschang, Knots, de Gruyter Studies in Math. 5, Walter de Gruyter (2003)

[3] S E Cappell, J L Shaneson, Invariants of $3$–manifolds, Bull. Amer. Math. Soc. 81 (1975) 559

[4] S E Cappell, J L Shaneson, Linking numbers in branched covers, from: "Four-manifold theory (Durham, N.H., 1982)" (editors C Gordon, R Kirby), Contemp. Math. 35, Amer. Math. Soc. (1984) 165

[5] T D Cochran, A Gerges, K Orr, Dehn surgery equivalence relations on $3$–manifolds, Math. Proc. Cambridge Philos. Soc. 131 (2001) 97

[6] D Cooper, C D Hodgson, S P Kerckhoff, Three-dimensional orbifolds and cone-manifolds, MSJ Memoirs 5, Math. Soc. Japan (2000)

[7] R H Fox, A quick trip through knot theory, from: "Topology of $3$–manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961)", Prentice-Hall (1962) 120

[8] S Garoufalidis, A Kricker, A surgery view of boundary links, Math. Ann. 327 (2003) 103

[9] S Garoufalidis, A Kricker, Finite type invariants of cyclic branched covers, Topology 43 (2004) 1247

[10] S Garoufalidis, A Kricker, A rational noncommutative invariant of boundary links, Geom. Topol. 8 (2004) 115

[11] P M Gilmer, Configurations of surfaces in $4$–manifolds, Trans. Amer. Math. Soc. 264 (1981) 353

[12] P M Gilmer, Classical knot and link concordance, Comment. Math. Helv. 68 (1993) 1

[13] K Habiro, Claspers and finite type invariants of links, Geom. Topol. 4 (2000) 1

[14] R A Litherland, S D Wallace, Surgery description of colored knots, Algebr. Geom. Topol. 8 (2008) 1295

[15] S V Matveev, Generalized surgeries of three-dimensional manifolds and representations of homology spheres, Mat. Zametki 42 (1987) 268, 345

[16] D Moskovich, Surgery untying of coloured knots, Algebr. Geom. Topol. 6 (2006) 673

[17] H Murakami, Y Nakanishi, On a certain move generating link-homology, Math. Ann. 284 (1989) 75

[18] S Naik, T Stanford, A move on diagrams that generates $S$–equivalence of knots, J. Knot Theory Ramifications 12 (2003) 717

[19] K A Perko Jr., On the classification of knots, Proc. Amer. Math. Soc. 45 (1974) 262

[20] J H Przytycki, M V Sokolov, Surgeries on periodic links and homology of periodic $3$–manifolds, Math. Proc. Cambridge Philos. Soc. 131 (2001) 295

[21] K Reidemeister, Knoten und Verkettungen, Math. Z. 29 (1929) 713

[22] D Rolfsen, Knots and links, Math. Lecture Series 7, Publish or Perish (1990)

[23] J J Rotman, An introduction to the theory of groups, Graduate Texts in Math. 148, Springer (1995)

[24] C P Rourke, B J Sanderson, Introduction to piecewise-linear topology, Ergebnisse der Math. und ihrer Grenzgebiete 69, Springer (1972)

[25] M Sakuma, Surgery description of orientation-preserving periodic maps on compact orientable $3$–manfolds, Rend. Istit. Mat. Univ. Trieste 32 (2001)

[26] H Seifert, W Threlfall, A textbook of topology, Pure and Applied Math. 89, Academic Press (1980)

[27] F J Swenton, Algorithmic construction of Kirby diagrams for branched covers, J. Knot Theory Ramifications 13 (2004) 939

[28] T Yamada, Translation algorithms between the covering presentation and other presentations of $3$–manifolds, dissertation, Tokyo Institute of Technology (2002)