The Michigan Mathematical Journal

Extensions of Some Classical Local Moves on Knot Diagrams

Benjamin Audoux, Paolo Bellingeri, Jean-Baptiste Meilhan, and Emmanuel Wagner

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Abstract

We consider local moves on classical and welded diagrams: (self-)crossing change, (self-)virtualization, virtual conjugation, Delta, fused, band-pass, and welded band-pass moves. Interrelationships between these moves are discussed, and, for each of these moves, we provide an algebraic classification. We address the question of relevant welded extensions for classical moves in the sense that the classical quotient of classical object embeds into the welded quotient of welded objects. As a byproduct, we obtain that all of the local moves mentioned are unknotting operations for welded (long) knots. We also mention some topological interpretations for these combinatorial quotients.

Article information

Source
Michigan Math. J., Volume 67, Issue 3 (2018), 647-672.

Dates
Received: 16 December 2016
Revised: 20 June 2017
First available in Project Euclid: 13 July 2018

Permanent link to this document
https://projecteuclid.org/euclid.mmj/1531447373

Digital Object Identifier
doi:10.1307/mmj/1531447373

Mathematical Reviews number (MathSciNet)
MR3835567

Zentralblatt MATH identifier
06969987

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M27: Invariants of knots and 3-manifolds 20F36: Braid groups; Artin groups

Citation

Audoux, Benjamin; Bellingeri, Paolo; Meilhan, Jean-Baptiste; Wagner, Emmanuel. Extensions of Some Classical Local Moves on Knot Diagrams. Michigan Math. J. 67 (2018), no. 3, 647--672. doi:10.1307/mmj/1531447373. https://projecteuclid.org/euclid.mmj/1531447373


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References

  • [1] H. Aida, The oriented ${\Delta}_{ij}$-moves on links, Kobe J. Math. 9 (1992), no. 2, 163–170.
  • [2] B. Audoux, On the welded Tube map, Contemp. Math. 670 (2016), 261–284.
  • [3] B. Audoux, P. Bellingeri, J.-B. Meilhan, and E. Wagner, Homotopy classification of ribbon tubes and welded string links, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 17 (2017), no. 2, 713–761.
  • [4] B. Audoux, P. Bellingeri, J.-B. Meilhan, and E. Wagner, On usual, virtual and welded knotted objects up to homotopy, J. Math. Soc. Japan, 69 (2017), no. 3, 1079–1097.
  • [5] B. Audoux, J.-B. Meilhan, and E. Wagner, On codimension two embeddings up to link-homotopy, J. Topol., 10 (2017), no. 4, 1107–1123.
  • [6] D. Bar-Natan and Z. Dancso, Finite-type invariants of w-knotted objects, I: w-knots and the Alexander polynomial, Algebr. Geom. Topol. 16 (2016), no. 2, 1063–1133.
  • [7] V. G. Bardakov, P. Bellingeri, and C. Damiani, Unrestricted virtual braids, fused links and other quotients of virtual braid groups, J. Knot Theory Ramifications 24 (2015), no. 12, 1550063.
  • [8] T. E. Brendle and A. Hatcher, Configuration spaces of rings and wickets, Comment. Math. Helv. 88 (2013), no. 1, 131–162.
  • [9] J.S. Carter, S. Kamada, and M. Saito, Stable equivalence of knots on surfaces and virtual knot cobordisms, J. Knot Theory Ramifications 11 (2002), no. 3, 311–322; Knots 2000 Korea, Vol. 1 (Yongpyong).
  • [10] J.S. Carter, S. Kamada, M. Saito, and S. Satoh, A theorem of Sanderson on link bordisms in dimension $4$, Algebr. Geom. Topol. 1 (2001), 299–310 (electronic).
  • [11] B. A. Cisneros De La Cruz, Virtual braids from a topological viewpoint, J. Knot Theory Ramifications 24 (2015), no. 6, 1550033.
  • [12] C. Damiani, A journey through loop braid groups, Expo. Math., 35, (2017), 252–285.
  • [13] R. Fenn, R. Rimányi, and C. Rourke, The braid-permutation group, Topology 36 (1997), no. 1, 123–135.
  • [14] T. Fiedler, Gauss diagram invariants for knots and links, Math. Appl., 532, Kluwer Academic Publishers, Dordrecht, 2001.
  • [15] A. Fish and E. Keyman, Classifying links under fused isotopy, J. Knot Theory Ramifications 25 (2016), no. 7, 1650042.
  • [16] M. Goussarov, M. Polyak, and O. Viro, Finite-type invariants of classical and virtual knots, Topology 39 (2000), no. 5, 1045–1068.
  • [17] N. Habegger and X.-S. Lin, The classification of links up to link-homotopy, J. Amer. Math. Soc. 3 (1990), 389–419.
  • [18] L. H. Kauffman, Virtual knot theory, European J. Combin. 20 (1999), no. 7, 663–690.
  • [19] L. H. Kauffman, A survey of virtual knot theory, Knots in Hellas ’98 (Delphi), Ser. Knots Everything, 24, pp. 143–202, World Sci. Publ., River Edge, NJ, 2000.
  • [20] L. H. Kauffman and S. Lambropoulou, Virtual braids, Fund. Math. 184 (2004), 159–186.
  • [21] L. H. Kauffman and S. Lambropoulou, Virtual braids and the $L$-move, J. Knot Theory Ramifications 15 (2006), no. 6, 773–811.
  • [22] A. Kawauchi, A survey of knot theory, Birkhäuser, Basel, 1996.
  • [23] S. V. Matveev, Generalized surgeries of three-dimensional manifolds and representations of homology spheres, Mat. Zametki 42 (1987), no. 2, 268–278, 345.
  • [24] J.-B. Meilhan, On Vassiliev invariants of order two for string links, J. Knot Theory Ramifications 14 (2005), no. 5, 665–687.
  • [25] J. Milnor, Link groups, Ann. of Math. (2) 59 (1954), 177–195.
  • [26] H. Murakami, Some metrics on classical knots, Math. Ann. 270 (1985), 35–45.
  • [27] H. Murakami and Y. Nakanishi, On a certain move generating link-homology, Math. Ann. 284 (1989), no. 1, 75–89.
  • [28] K. Murasugi and B. I. Kurpita, A study of braids, Math. Appl., 484, Kluwer Academic Publishers, Dordrecht, 1999.
  • [29] Y. Nakanishi, Fox’s congruence modulo $(2,1)$, Sûrikaisekikenkyûsho Kôkyûroku 813 (1985), 102–110.
  • [30] Y. Nakanishi, Replacements in the Conway third identity, Tokyo J. Math. 14 (1991), no. 1, 197–203.
  • [31] T. Nasybullov, The classification of fused links, J. Knot Theory Ramifications 25 (2016), no. 21, 1650076.
  • [32] K. Reidemeister, Unveränderter reprografischer, Einführung in die kombinatorische Topologie, Nachdr. Ausg. Braunschw., Wissenschaftliche Buchgesellschaft, Darmstadt, 1972. Unveränderter reprografischer Nachdruck der Ausgabe Braunschweig 1951.
  • [33] B. J. Sanderson, Bordism of links in codimension $2$, J. Lond. Math. Soc. (2) 35 (1987), no. 2, 367–376.
  • [34] B. J. Sanderson, Triple links in codimension $2$, Topology. Theory and applications, II (Pécs, 1989), Colloq. Math. Soc. János Bolyai, 55, pp. 457–471, North-Holland, Amsterdam, 1993.
  • [35] S. Satoh, Virtual knot presentation of ribbon torus-knots, J. Knot Theory Ramifications 9 (2000), no. 4, 531–542.
  • [36] S. Satoh, Crossing changes, delta moves and sharp moves on welded knots, Rocky Mountain J. Math. (2015, to appear), arXiv:1510.02554.
  • [37] B. Winter, The classification of spun torus knots, J. Knot Theory Ramifications 18 (2009), no. 9, 1287–1298.
  • [38] T. Yajima, On the fundamental groups of knotted $2$-manifolds in the $4$-space, J. Math., Osaka City Univ. 13 (1962), 63–71.