View source for Montesinos-Nakanishi conjecture

Any [[Link|link]] can be reduced to a trivial link by a sequence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130210/m1302102.png" />-moves (that is, moves which add three half-twists into two parallel arcs of a link).

The conjecture has been proved for links up to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130210/m1302103.png" /> crossings, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130210/m1302104.png" />-bridge links and five-braid links except one family represented by the square of the centre of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130210/m1302105.png" />-braid group. This link, which can be reduced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130210/m1302106.png" />-moves to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130210/m1302107.png" />-crossings link, is the smallest known link for which the conjecture is open (as of 2001).

The conjecture has its stronger version that any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130210/m1302109.png" />-tangle can be reduced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130210/m13021010.png" />-moves to one of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130210/m13021011.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130210/m13021012.png" />-tangles (with possible additional trivial components), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130210/m13021013.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Kirby,   "Problems in low-dimensional topology"  W. Kazez (ed.) , ''Geometric Topology (Proc. Georgia Internat. Topol. Conf. 1993)'' , ''Stud. Adv. Math.'' , '''2:2''' , Amer. Math. Soc. /IP  (1997)  pp. 35–473</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  Q. Chen,   "The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130210/m13021014.png" />-move conjecture for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130210/m13021015.png" />-braids" , ''Knots in Hellas '98 (Proc. Internat. Conf. Knot Theory and Its Ramifications'' , ''Knots and Everything'' , '''24'''  (2000)  pp. 36–47</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.H. Przytycki,   T. Tsukamoto,   "The fourth skein module and the Montesinos–Nakanishi conjecture for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130210/m13021016.png" />-algebraic links"  ''J. Knot Th. Ramifications'' , '''to appear'''  (2001)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H.R. Morton,   "Problems"  J.S. Birman (ed.)  A. Libgober (ed.) , ''Braids (Santa Cruz, 1986)'' , ''Contemp. Math.'' , '''78''' , Amer. Math. Soc.  (1988)  pp. 557–574</TD></TR></table>