Journal of the Mathematical Society of Japan

On usual, virtual and welded knotted objects up to homotopy

Abstract

We consider several classes of knotted objects, namely usual, virtual and welded pure braids and string links, and two equivalence relations on those objects, induced by either self-crossing changes or self-virtualizations. We provide a number of results which point out the differences between these various notions. The proofs are mainly based on the techniques of Gauss diagram formulae.

Article information

Source
J. Math. Soc. Japan, Volume 69, Number 3 (2017), 1079-1097.

Dates
First available in Project Euclid: 12 July 2017

https://projecteuclid.org/euclid.jmsj/1499846518

Digital Object Identifier
doi:10.2969/jmsj/06931079

Mathematical Reviews number (MathSciNet)
MR3685036

Zentralblatt MATH identifier
06786989

Citation

AUDOUX, Benjamin; BELLINGERI, Paolo; MEILHAN, Jean-Baptiste; WAGNER, Emmanuel. On usual, virtual and welded knotted objects up to homotopy. J. Math. Soc. Japan 69 (2017), no. 3, 1079--1097. doi:10.2969/jmsj/06931079. https://projecteuclid.org/euclid.jmsj/1499846518

References

• E. Artin, Theorie der zöpfe, Abh. Math. Semin. Univ. Hamb., 4 (1926), 47–72.
• B. Audoux, P. Bellingeri, J.-B. Meilhan and E. Wagner, Homotopy classification of ribbon tubes and welded string links, arXiv e-prints:1407.0184, 2014, to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), XVII (2017), 713–761.
• D. Bar-Natan, Balloons and hoops and their universal finite-type invariant, BF theory, and an ultimate Alexander invariant, Acta Math. Vietnam., 40 (2015), 271–329.
• D. Bar-Natan and Z. Dancso, Finite-type invariants of w-knotted objects, II: Tangles, foams and the kashiwara-vergne, Math. Ann., 367 (2017), 1517–1586.
• 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), 1063–1133.
• V. G. Bardakov, The virtual and universal braids, Fund. Math., 184 (2004), 1–18.
• T. E. Brendle and A. Hatcher, Configuration spaces of rings and wickets, Comment. Math. Helv., 88 (2013), 131–162.
• J. Carter, S. Kamada and M. Saito, Stable equivalence of knots on surfaces and virtual knot cobordisms, J. Knot Theory Ramifications, 11 (2002), 311–322.
• Z. Cheng and H. Gao, A polynomial invariant of virtual links, J. Knot Theory Ramifications, 22, no.12, 2013.
• H. A. Dye and L. H. Kauffman, Virtual homotopy, J. Knot Theory Ramifications, 19 (2010), 935–960.
• R. Fenn, R. Rimányi and C. Rourke, The braid-permutation group, Topology, 36 (1997), 123–135.
• T. Fiedler, Gauss diagram invariants for knots and links, Math. Appl., 532, Kluwer Academic Publishers, Dordrecht, 2001.
• D. L. Goldsmith, Homotopy of braids - in answer to a question of E. Artin, Topology Conf., Virginia Polytechnic Inst. and State Univ. 1973, Lect. Notes in Math., 375 (1974), 91–96.
• M. Goussarov, M. Polyak and O. Viro, Finite type invariants of virtual and classical knots, Topology, 39 (2000), 1045–1168.
• N. Habegger and X.-S. Lin, The classification of links up to link-homotopy, J. Amer. Math. Soc., 3 (1990), 389–419.
• N. Habegger and X.-S. Lin, On link concordance and Milnor's $\overline {\mu}$ invariants, Bull. London Math. Soc., 30 (1998), 419–428.
• A. Ichimori and T. Kanenobu, Ribbon torus knots presented by virtual knots with up to four crossings, J. Knot Theory Ramifications, 21 (2012), 1240005, 30pp.
• L. H. Kauffman, Virtual knot theory, European J. Combin., 20 (1999), 663–690.
• G. Kuperberg, What is a virtual link? Algebr. Geom. Topol., 3 (2003), 587–591.
• J.-B. Meilhan, On Vassiliev invariants of order two for string links, J. Knot Theory Ramifications, 14 (2005), 665–687.
• J. Milnor, Link groups, Ann. of Math. (2), 59 (1954), 177–195.
• A. Mortier, Polyak type equations for virtual arrow diagram invariants in the annulus, J. Knot Theory Ramifications, 22 (2013), 1350034, 21pp.
• M. Polyak, On the algebra of arrow diagrams, Lett. Math. Phys., 51 (2000), 275–291.
• M. Polyak and O. Viro, Gauss diagram formulas for Vassiliev invariants, Internat. Math. Res. Notices, no.11 (1994), 445ff., 8pp.
• S. Satoh, Virtual knot presentation of ribbon torus-knots, J. Knot Theory Ramifications, 9 (2000), 531–542.
• T. Yajima, On the fundamental groups of knotted $2$-manifolds in the $4$-space, J. Math. Osaka City Univ., 13 (1962), 63–71.