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A LOWER BOUND FOR THE NUMBER OF FORBIDDEN MOVES TO UNKNOT A LONG VIRTUAL KNOT

MYEONG-JU JEONG

Department of Mathematics, Korea Science Academy of KAIST, Busan 614-822, Republic of Korea

https://doi.org/10.1142/S0218216513500247Cited by:1 (Source: Crossref)

Abstract

Nelson and Kanenobu showed that forbidden moves unknot any virtual knot. Similarly a long virtual knot can be unknotted by a finite sequence of forbidden moves. Goussarov, Polyak and Viro introduced finite type invariants of virtual knots and long virtual knots and gave combinatorial representations of finite type invariants. We introduce Fⁿ-moves which generalize the forbidden moves. Assume that two long virtual knots K and K′ are related by a finite sequence of Fⁿ-moves. We show that the values of the finite type invariants of degree 2 of K and K′ are congruent modulo n and give a lower bound for the number of Fⁿ-moves needed to transform K to K′.

Keywords:

AMSC: 57M25, 57M27

References

J. S. Birman and X.-S. Lin, Invent. Math. 111, 225 (1993), DOI: 10.1007/BF01231287. Crossref, Web of Science, Google Scholar
D. Bar-Natan, Topology 34, 423 (1995), DOI: 10.1016/0040-9383(95)93237-2. Crossref, Google Scholar
J. Conant, J. Mostovoy and T. Stanford, J. Knot Theory Ramifications 19(3), 355 (2010), DOI: 10.1142/S0218216510007875. Link, Web of Science, Google Scholar
M. Goussarov, M. Polyak and O. Viro, Topology 39, 1045 (2000), DOI: 10.1016/S0040-9383(99)00054-3. Crossref, Web of Science, Google Scholar
S. Kamada, Braid presentation of virtual knots and welded knots, preprint . Google Scholar
T. Kanenobu, J. Knot Theory Ramifications 10(1), 89 (2001), DOI: 10.1142/S0218216501000731. Link, Web of Science, Google Scholar
L. H. Kauffman, European J. Combin. 20, 663 (1999), DOI: 10.1006/eujc.1999.0314. Crossref, Web of Science, Google Scholar
H. Murakami and Y. Nakanishi, Math. Ann. 284, 75 (1989), DOI: 10.1007/BF01443506. Crossref, Web of Science, Google Scholar
Y. Nakanishi, J. Knot Theory Ramifications 2, 197 (1994). Google Scholar
S. Nelson, J. Knot Theory Ramifications 10, 931 (2001), DOI: 10.1142/S0218216501001244. Link, Web of Science, Google Scholar
Y. Ohyama and H. Yamada, J. Knot Theory Ramifications 17(7), 771 (2008), DOI: 10.1142/S0218216508006403. Link, Web of Science, Google Scholar
M. Polyak and O. Viro, Int. Math. Res. Notices 11, 445 (1994). Google Scholar
V. A. Vassiliev , Theory of Singularities and Its Applications , Advances in Soviet Mathematics 1 , ed. V. I. Arnold ( American Mathematical Society , 1990 ) . Crossref, Google Scholar