Research ArticleNo Access

Embeddings from the set of long flat virtual knots to the set of long virtual knots and applications

Young Ho Im

Department of Mathematics, Pusan National University, Busan, 609-735, Korea

E-mail Address: yhim@pusan.ac.kr

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Sera Kim

Department of Mathematics, Pusan National University, Busan, 609-735, Korea

E-mail Address: srkim@pusan.ac.kr

Corresponding author.

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, and

Kyeonghui Lee

Department of Mathematics, Pusan National University, Busan, 609-735, Korea

E-mail Address: lkh0202@pusan.ac.kr

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https://doi.org/10.1142/S021821652050025XCited by:0 (Source: Crossref)

Abstract

We investigate embeddings from the set of long flat virtual knot diagrams to the set of long virtual knot diagrams so that we can construct invariants for long flat virtual knots. Also, we give properties and examples of several invariants for long flat virtual knots via these embeddings and invariants for long virtual knots.

Keywords:

AMSC: 57M25, 57M27

References

1. M. Brandenbursky, Link invariants via counting surfaces, Geom. Dedicata 173 (2014) 243–270. Crossref, Web of Science, Google Scholar
2. S. Chomutov, S. Duzhin and J. Mostovoy, Introduction to Vassiliev knot invariants (Cambridge University Press, Cambridge, 2012). Crossref, Google Scholar
3. H. A. Dye and L. H. Kauffman, Virtual crossing number and the arrow polynomial, J. Knot Theory Ramifications 18(10) (2009) 1335–1357. Link, Web of Science, Google Scholar
4. N. Gügümcü and L. H. Kauffman, New invariants of knotoids, European J. Combin. 65 (2017) 186–229. Crossref, Web of Science, Google Scholar
5. D. Hrencecin and L. Kauffman, On filamentations and virtual knots, Topol. Appl. 134(1) (2003) 23–52. Crossref, Web of Science, Google Scholar
6. Y. H. Im and S. Kim, A sequence of polynomial invariants for Gauss diagrams, J. Knot Theory Ramifications 26(7) (2017) 1750039. Link, Web of Science, Google Scholar
7. Y. H. Im and K. Lee, A polynomial invariant of long virtual knots, European J. Combin. 30(5) (2009) 1289–1296. Crossref, Web of Science, Google Scholar
8. Y. H. Im and S. Y. Lee, A four-variable index polynomial invariant of long virtual knots, J. Knot Theory Ramifications 21(9) (2012) 1250083, 18 pp. Link, Web of Science, Google Scholar
9. L. H. Kauffman, Virtual knot theory, European J. Combin. 20(7) (1999) 663–690. Crossref, Web of Science, Google Scholar
10. L. H. Kauffman, Introduction to virtual knot theory, J. Knot theory Ramifications 21(13) (2012) 1240007, 37 pp. Link, Web of Science, Google Scholar
11. L. Kauffman and B. Richter, A polynomial invariant for flat virtual links, J. Knot theory Ramifications 17(5) (2008) 521–528. Link, Web of Science, Google Scholar
12. O. Östlund, Invariants of knot diagrams and relations among Reidemeister moves, J. Knot Theory Ramifications 10(8) (2001) 1215–1227. Link, Web of Science, Google Scholar
13. M. Polyak, Minimal generating sets of Reidemeister moves, Quantum Topol. 1 (2010) 399–411. Crossref, Google Scholar
14. V. Turaev, Virtual strings, Ann. Inst. Fourier (Grenoble) 54 (2004) 2455–2525. Crossref, Web of Science, Google Scholar