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Hass, Joel; Nowik, Tahl

Unknot diagrams requiring a quadratic number of Reidemeister moves to untangle. (English) Zbl 1191.57006

Discrete Comput. Geom. 44, No. 1, 91-95 (2010).
Summary: Given any knot diagram \(E\), we present a sequence of knot diagrams of the same knot type for which the minimum number of Reidemeister moves required to pass to \(E\) is quadratic with respect to the number of crossings. These bounds apply both in \(S ^{2}\) and in \(\mathbb R^{2}\).

Cited in 1 Review
Cited in 12 Documents

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)

Keywords:

Reidemister moves; unknot
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References:

[1] Alexander, J.W., Briggs, G.B.: On types of knotted curves. Ann. Math. 28, 562–586 (1926/1927) · JFM 53.0549.02 · doi:10.2307/1968399
[2] Carter, J.S., Elhamdadi, M., Saito, M., Satoh, S.: A lower bound for the number of Reidemeister moves of type III. Topol. Its Appl. 153, 2788–2794 (2006) · Zbl 1106.57003 · doi:10.1016/j.topol.2005.11.011
[3] Hagge, T.J.: Every Reidemeister move is needed for each knot type. Proc. Am. Math. Soc. 134(1), 295–301 (2006) · Zbl 1078.57007 · doi:10.1090/S0002-9939-05-07935-9
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[6] Hass, J., Snoeyink, J., Thurston, W.P.: The size of spanning disks for polygonal curves. Discrete Comput. Geom. 29(1), 1–17 (2003) · Zbl 1015.57008
[7] Hayashi, C.: A lower bound for the number of Reidemeister moves for unknotting. J. Knot Theory Ramif. 15(3), 313–325 (2006) · Zbl 1094.57007 · doi:10.1142/S0218216506004488
[8] Östlund, O.: Invariants of knot diagrams and relations among Reidemeister moves. J. Knot Theory Ramif. 10(8), 1215–1227 (2001) · Zbl 0998.57021 · doi:10.1142/S0218216501001402
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