Braid equivalences in 3-manifolds with rational surgery description. (English) Zbl 1353.57015
This paper gives mixed braid equivalences from both geometric and algebraic viewpoints.
It is well-known that any closed connected orientable \(3\)-manifold, say \(M\), is obtained by Dehn surgery on a link, say \(K_0\), in the \(3\)-sphere. Hence a link \(K\) in \(M\) can be represented by a mixed link in the \(3\)-sphere which is the union of \(K\) and \(K_0\) with surgery coefficients. Its braid presentation is called a mixed braid. If \(M\) is obtained by integral surgery on \(K_0\), then such equivalences have already been obtained by the second author et al. (cf. [Topology Appl. 78, No. 1–2, 95–122 (1997; Zbl 0879.57007); Compos. Math. 142, No. 4, 1039–1062 (2006; Zbl 1156.57007)]).
The authors, in this paper, generalize the results to the case that \(M\) is obtained by rational surgery on \(K_0\) in the \(3\)-sphere. The first main theorem, which is given from the geometric viewpoint, states that two links in \(M\) are isotopic if and only if their mixed braids can be transformed into each other by a finite sequence of two moves: \(L\)-moves and band moves. The second main theorem, which is in turn given from the algebraic viewpoint, states that two links in \(M\) are isotopic if and only if their mixed braid representatives in the “braid group” are equivalent up to four moves: \(M\)-moves, \(M\)-conjugation, combed loop conjugation and combed algebraic braid band moves.
The authors end this paper by mentioning that the results contain a lot of applications: to the study of Jones type invariants for links in \(M\), to the study of skein modules of \(M\) and to a computational approach to the Witten invariants.
It is well-known that any closed connected orientable \(3\)-manifold, say \(M\), is obtained by Dehn surgery on a link, say \(K_0\), in the \(3\)-sphere. Hence a link \(K\) in \(M\) can be represented by a mixed link in the \(3\)-sphere which is the union of \(K\) and \(K_0\) with surgery coefficients. Its braid presentation is called a mixed braid. If \(M\) is obtained by integral surgery on \(K_0\), then such equivalences have already been obtained by the second author et al. (cf. [Topology Appl. 78, No. 1–2, 95–122 (1997; Zbl 0879.57007); Compos. Math. 142, No. 4, 1039–1062 (2006; Zbl 1156.57007)]).
The authors, in this paper, generalize the results to the case that \(M\) is obtained by rational surgery on \(K_0\) in the \(3\)-sphere. The first main theorem, which is given from the geometric viewpoint, states that two links in \(M\) are isotopic if and only if their mixed braids can be transformed into each other by a finite sequence of two moves: \(L\)-moves and band moves. The second main theorem, which is in turn given from the algebraic viewpoint, states that two links in \(M\) are isotopic if and only if their mixed braid representatives in the “braid group” are equivalent up to four moves: \(M\)-moves, \(M\)-conjugation, combed loop conjugation and combed algebraic braid band moves.
The authors end this paper by mentioning that the results contain a lot of applications: to the study of Jones type invariants for links in \(M\), to the study of skein modules of \(M\) and to a computational approach to the Witten invariants.
Reviewer: Toshio Saito (Joetsu)
MSC:
| 57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |
| 57M25 | Knots and links in the \(3\)-sphere (MSC2010) |
| 57N10 | Topology of general \(3\)-manifolds (MSC2010) |
Keywords:
rational surgery; links in 3-manifolds; mixed links; band moves; mixed braids; \(L\)-moves; Markov moves; braid band moves; parting; combing; cabling; lens spaces; Seifert manifolds; homology spheresReferences:
| [3] | Diamantis, I.; Lambropoulou, S., A new basis for the Homflypt skein module of the solid torus (2014) |
| [5] | Gabrovsek, B.; Mroczkowski, M., The HOMFLYPT skein module of the lens spaces \(L(p, 1)\), Topol. Appl., 175, 72-80 (2014) · Zbl 1298.57012 |
| [6] | Hoste, J.; Kidwell, M., Dichromatic link invariants, Trans. Am. Math. Soc., 321, 1, 197-229 (1990) · Zbl 0702.57002 |
| [7] | Häring-Oldenburg, R.; Lambropoulou, S., Knot theory in handlebodies, J. Knot Theory Ramif., 11, 6, 921-943 (2002) · Zbl 1023.57009 |
| [9] | Ko, K. H.; Smolinsky, L., The framed braid group and 3-manifolds, Proc. Am. Math. Soc., 115, 2, 541-551 (1992) · Zbl 0760.57007 |
| [10] | Lambropoulou, S., Braid structures in handlebodies, knot complements and 3-manifolds, (Proceedings of Knots in Hellas ’98. Proceedings of Knots in Hellas ’98, Series of Knots and Everything, vol. 24 (2000), World Scientific Press), 274-289 · Zbl 0976.57007 |
| [11] | Lambropoulou, S., Knot theory related to generalized and cyclotomic Hecke algebras of type B, J. Knot Theory Ramif., 8, 5, 621-658 (1999) · Zbl 0933.57007 |
| [12] | Lambropoulou, S.; Rourke, C. P., Markov’s theorem in 3-manifolds, Topol. Appl., 78, 95-122 (1997) · Zbl 0879.57007 |
| [13] | Lambropoulou, S.; Rourke, C. P., Algebraic Markov equivalence for links in 3-manifolds, Compos. Math., 142, 1039-1062 (2006) · Zbl 1156.57007 |
| [14] | Moser, L., Elementary surgery along a torus knot, Pac. J. Math., 38, 737-745 (1971) · Zbl 0202.54701 |
| [15] | Rolfsen, D., Knots and Links (1976), Publish or Perish, Inc.: Publish or Perish, Inc. Berkeley, CA · Zbl 0339.55004 |
| [16] | Rolfsen, D., Rational surgery calculus: extension of Kirby’s theorem, Pac. J. Math., 110, 2, 377-386 (1984) · Zbl 0488.57002 |
| [17] | Saveliev, N., Lectures on the Topology of 3-Manifolds. An Introduction to the Casson Invariant, de Gruyter Textbook (1999), de Gruyter: de Gruyter Berlin · Zbl 0932.57001 |
| [18] | Skora, R., Knot and link projections in 3-manifolds, Math. Z., 206, 345-350 (1991) · Zbl 0695.57006 |
| [19] | Skora, R., Closed braids in 3-manifolds, Math. Z., 211, 173-187 (1992) · Zbl 0758.57007 |
| [20] | Sossinsky, A. B., Preparation theorems for isotopy invariants of links in 3-manifolds, (Quantum Groups, Proceedings. Quantum Groups, Proceedings, Lecture Notes in Math., vol. 1510 (1992), Springer-Verlag: Springer-Verlag Berlin), 354-362 · Zbl 0765.57009 |
| [21] | Sundheim, P. A., Reidemeister’s theorem for 3-manifolds, Math. Proc. Camb. Philos. Soc., 110, 281-292 (1991) · Zbl 0754.57012 |
| [22] | Sundheim, P. A., The Alexander and Markov theorems via diagrams for links in 3-manifolds, Trans. Am. Math. Soc., 337, 2, 591-607 (1993) · Zbl 0795.57007 |
| [23] | Turaev, V. G., The Conway and Kauffman modules of the solid torus, Zap. Nauč. Semin. POMI, 167, 79-89 (1988), English translation: J. Soviet Math. (1990), 2799-2805 · Zbl 0673.57004 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
