Hostname: page-component-cb9f654ff-nr592 Total loading time: 0 Render date: 2025-08-18T03:59:40.716Z Has data issue: false hasContentIssue false
  • English
  • Français

The Chord Index, its Definitions, Applications, and Generalizations

Part of: Low-dimensional topology

Published online by Cambridge University Press:  30 January 2020

Zhiyun Cheng
Zhiyun Cheng*
Affiliation:
School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Beijing Normal University, Beijing 100875, China
*
e-mail: czy@bnu.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we study the chord index of virtual knots, which can be thought of as an extension of the chord parity. We show how to use the chord index to enhance the quandle coloring invariants. The notion of indexed quandle is introduced, which generalizes the quandle idea. Some applications of this new invariant is discussed. We also study how to define a generalized chord index via a fixed finite biquandle. Finally, the chord index and its applications in twisted knot theory are discussed.

Keywords

virtual knotchord indexwrithe polynomialindexed quandletwisted knot

MSC classification

Primary: 57M25: Knots and links in $S^3$
Secondary: 57M27: Invariants of knots and 3-manifolds

Information

Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 3 , June 2021 , pp. 597 - 621
Check for updates
Copyright
© Canadian Mathematical Society 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

The author is supported by NSFC 11771042, NSFC 11571038 and China Scholarship Council.

References

Bourgoin, M. O., Twisted link theory. Algebr. and Geom. Topol. 8(2008), 1249–1279. https://doi.org/10.2140/agt.2008.8.1249 CrossRefGoogle Scholar
Carter, J. S., A survey of quandle ideas. In: Introductory lectures on knot theory, Ser. Knots Everything, 46, World Sci. Publ., Hackensack, NJ, 2012, pp. 22–53. https://doi.org/10.1142/9789814313001_0002 CrossRefGoogle Scholar
Carter, J. S., Elhamdadi, M., Nikiforou, M. A., Saito, M., Extensions of quandles and cocycle knot invariants. J. Knot Theory Ramifications 12(2003), 725–738. https://doi.org/10.1142/S0218216503002718 CrossRefGoogle Scholar
Carter, J. S., Jelsovsky, D., Kamada, S., Langford, L., and Saito, M., Quandle cohomology and state-sum invariants of knotted curves and surfaces. Trans. Amer. Math. Soc., 355(2003), 3947–3989. https://doi.org/10.1090/S0002-9947-03-03046-0 CrossRefGoogle Scholar
Carter, J. S., Kamada, S., and Saito, M., Stable equivalence of knots on surfaces and virtual knot cobordisms. J. Knot Theory Ramifications 11(2002), 311–322. https://doi.org/10.1142/S0218216502001639 CrossRefGoogle Scholar
Cheng, Z., A polynomial invariant of virtual knots. Proc. Amer. Math. Soc. 142(2014), 713–725. https://doi.org/10.1090/S0002-9939-2013011785-5 CrossRefGoogle Scholar
Cheng, Z., A transcendental function invariant of virtual knots. J. Math. Soc. Japan 69(2017), 1583–1599. https://doi.org/10.2969/jmsj/06941583 CrossRefGoogle Scholar
Cheng, Z. and Gao, H., A polynomial invariant of virtual links. J. Knot Theory Ramifications 22(2013), 1341002. https://doi.org/10.1142/S0218216513410022 CrossRefGoogle Scholar
Cheng, Z., Gao, H., and Xu, M., Some remarks on the chord index. arXiv:1811.09061Google Scholar
Chrisman, M. W. and Dye, H. A., The three loop isotopy and framed isotopy invariants of virtual knots. Topology Appl. 173(2014), 107–134. https://doi.org/10.1016/j.topol.2014.05.011 CrossRefGoogle Scholar
Clark, W., Elhamdadi, M., Saito, M., and Yeatman, T., Quandle colorings of knots and applications. J. Knot Theory Ramifications 23(2014), 1450035. https://doi.org/10.1142/S0218216514500357 CrossRefGoogle ScholarPubMed
Dye, H. A., Vassiliev invariants from parity mappings. J. Knot Theory Ramifications 22(2013), 1340008. https://doi.org/10.1142/S0218216513400087 CrossRefGoogle Scholar
Dye, H. A. and Kauffman, L. H., Virtual crossing number and the arrow polynomial. J. Knot Theory Ramifications 18(2009), 1335–1357. https://doi.org/10.1142/S0218216509007166 CrossRefGoogle Scholar
Fenn, R., llyutko, D. P., Kauffman, L. H., and Manturov, V. O., Unsolved problems in virtual knot theory and combinatorial knot theory. Banach Center Publications 103(2014), 9–61. https://doi.org/10.4064/bc103-0-1 CrossRefGoogle Scholar
Fenn, R., Jordan-Santana, M., and Kauffman, L. H., Biquandles and virtual links. Topology Appl. 145(2004), 157–175. https://doi.org/10.1016/j.topol.2004.06.008 CrossRefGoogle Scholar
Folwaczny, L. C. and Kauffman, L. H., A linking number definition of the affine index polynomial and applications. J. Knot Theory Ramifications 22(2013), 1341004. https://doi.org/10.1142/S0218216513410046 CrossRefGoogle Scholar
Goussarov, M., Polyak, M., and Viro, O., Finite-type invariants of classical and virtual knots. Topology 39(2000), 1045–1068. https://doi.org/10.1016/S0040-9383(99)00054-3 CrossRefGoogle Scholar
Henrich, A., A Sequence of Degree One Vassiliev Invariants for Virtual Knots. J. Knot Theory Ramifications 19(2010), 461–487. https://doi.org/10.1142/S0218216510007917 CrossRefGoogle Scholar
Im, Y. H. and Kim, S., A sequence of polynomial invariants for Gauss diagrams. J. Knot Theory Ramifications 26(2017), 1750039. https://doi.org/10.1142/S0218216517500390 CrossRefGoogle Scholar
Im, Y. H., Kim, S., and Lee, D. S., The parity writhe polynomials for virtual knots and flat virtual knots. J. Knot Theory Ramifications 22(2013), 1250133. https://doi.org/10.1142/S0218216512501337 CrossRefGoogle Scholar
Joyce, D., A classifying invariant of knots, the knot quandle. J. Pure Appl. Algebra, 23(1982), 37–65. https://doi.org/10.1016/0022-4049(82)90077-9 CrossRefGoogle Scholar
Kaestner, A. M. and Kauffman, Louis H., Parity, skein polynomials and categorification. J. Knot Theory Ramifications 21(2012), 1240011. https://doi.org/10.1142/S0218216512400111 CrossRefGoogle Scholar
Kaestner, A. M. and Kauffman, L. H., Parity biquandles. In: Knots in Poland. III. Part 1, Banach Center Publ., 100, Polish Acad. Sci. Inst. Math., Warsaw, 2014, pp. 131–151. https://doi.org/10.4064/bc100-0-7 Google Scholar
Kaestner, A. M., Nelson, S., and Selker, L., Parity biquandle invariants of virtual knots. Topology Appl. 209(2016), 207–219. https://doi.org/10.1016/j.topol.2016.06.010 CrossRefGoogle Scholar
Kamada, N., Span of the Jones polynomial of an alternating virtual link. Algebr. Geom. Topol. 4(2004), 1083–1101. https://doi.org/10.2140/agt.2004.4.1083 CrossRefGoogle Scholar
Kamada, N., Polynomial invariants and quandles of twisted links. Topology Appl. 159(2012), 999–1006. https://doi.org/10.1016/j.topol.2011.11.024 CrossRefGoogle Scholar
Kamada, N., Index polynomial invariants of twisted links. J. Knot Theory Ramifications 22(2013), 1340005. https://doi.org/10.1142/S0218216513400051 CrossRefGoogle Scholar
H. Kauffman, L., State models and the Jones polynomial. Topology 26(1987), 395–407. https://doi.org/10.1016/0040-9383(87)90009-7 CrossRefGoogle Scholar
Kauffman, L. H., Virtual knot theory. European J. Combin. 20(1999), 663–691.CrossRefGoogle Scholar
Kauffman, L. H., A self-linking invariant of virtual knots. Fund. Math. 184(2004), 135–158. https://doi.org/10.4064/fm184-0-10 CrossRefGoogle Scholar
Kauffman, L. H., An affine index polynomial invariant of virtual knots. J. Knot Theory Ramifications 22(2013), 1340007. https://doi.org/10.1142/S0218216513400075 CrossRefGoogle Scholar
Kauffman, L. H. and Manturov, V. O., Virtual biquandles. Fund. Math. 188(2005), 103–146. https://doi.org/10.4064/fm188-0-6 CrossRefGoogle Scholar
Kontsevich, M., Vassiliev’s knot invariants. In: I. M. Gel’fand Seminar, Adv. Soviet Math. 16, Part 2, Amer. Math. Soc., Providence, RI, 1993, pp. 137–150.Google Scholar
Kuperberg, Greg, What is a virtual link? Algebr. Geom. Topol. 3(2003), 587–591. https://doi.org/10.2140/agt.2003.3.587 CrossRefGoogle Scholar
Manturov, V., Parity in knot theory. Sb. Math. 201(2010), 693–733. https://doi.org/10.1070/SM2010v201n05ABEH004089 CrossRefGoogle Scholar
Matveev, S. V., Distributive groupoids in knot theory. Math. USSR Sb., 47(1984), 73–83.CrossRefGoogle Scholar
Miyazawa, Y., Magnetic graphs and an invariant for virtual links. J. Knot Theory Ramifications 15(2006), 1319–1334. https://doi.org/10.1142/S0218216506005135 CrossRefGoogle Scholar
Miyazawa, Y., A multivariable polynomial invariant for unoriented virtual knots and links. J. Knot Theory Ramifications 17(2008), 1311–1326. https://doi.org/10.1142/S0218216508006658 CrossRefGoogle Scholar
Nakamura, T., Nakanishi, Y., and Satoh, S., A note on coverings of virtual knots. https://arXiv:1811.10852 Google Scholar
Nakamura, T., Nakanishi, Y., and Satoh, S., Writhe polynomials and shell moves for virtual knots and links. https://arXiv:1905.03489 Google Scholar
Rourke, C. and Sanderson, B., A new classification of links and some calculations using it. https://arXiv:math/0006062 Google Scholar
Satoh, S. and Taniguchi, K., The writhes of a virtual knot. Fund. Math. 225(2014), 327–342. https://doi.org/10.4064/fm225-1-15 CrossRefGoogle Scholar
Sawollek, J., On Alexander-Conway polynomials for virtual knots and links. https://arXiv:math/9912173v2 Google Scholar
Sawollek, J., An orientation-sensitive Vassiliev invariant for virtual knots, J. Knot Theory Ramifications 12(2003), 767–779. https://doi.org/10.1142/S0218216503002743 CrossRefGoogle Scholar
Silver, D. and Williams, S., Polynomial invariants of virtual links. J. Knot Theory Ramifications 12(2003), 987–1000. https://doi.org/10.1142/S0218216503002901 CrossRefGoogle Scholar
Turaev, V., Virtual strings. Ann Inst. Fourier (Grenoble) 54(2004), 2455–2525.CrossRefGoogle Scholar