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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).
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Any [[link]] can be reduced to a trivial link by a sequence of $3$-moves (that is, [[tangle move]]s 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).
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The conjecture has been proved for links up to $12$ crossings, $4$-bridge links and five-braid links except one family represented by the square of the centre of the $5$-braid group. This link, which can be reduced by $3$-moves to a $20$-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" />.
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The conjecture has its stronger version that any $n$-tangle can be reduced by $3$-moves to one of $g(n)$ $n$-tangles (with possible additional trivial components), where $g(n)=\prod_{i=1}^{n-1}(3^i+1)$.
  
 
====References====
 
====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>
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<table>
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<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 {{ZBL|0888.57014}}</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  Q. Chen,  "The $3$-move conjecture for $5$-braids" , ''Knots in Hellas '98 (Proc. Internat. Conf. Knot Theory and Its Ramifications'' , ''Knots and Everything'' , '''24'''  (2000)  pp. 36–47 {{ZBL|0973.57003}}</TD></TR>
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<TR><TD valign="top">[a3]</TD> <TD valign="top">  J.H. Przytycki,  T. Tsukamoto,  "The fourth skein module and the Montesinos–Nakanishi conjecture for $3$-algebraic links"  ''J. Knot Th. Ramifications'' , '''10'''  (2001) pp.959–982  {{DOI|10.1142/S0218216501001281}} {{ZBL|1001.57029}}</TD></TR>
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<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 {{ZBL|0968.57004}}</TD></TR>
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</table>

Latest revision as of 14:49, 31 December 2016

Any link can be reduced to a trivial link by a sequence of $3$-moves (that is, tangle moves which add three half-twists into two parallel arcs of a link).

The conjecture has been proved for links up to $12$ crossings, $4$-bridge links and five-braid links except one family represented by the square of the centre of the $5$-braid group. This link, which can be reduced by $3$-moves to a $20$-crossings link, is the smallest known link for which the conjecture is open (as of 2001).

The conjecture has its stronger version that any $n$-tangle can be reduced by $3$-moves to one of $g(n)$ $n$-tangles (with possible additional trivial components), where $g(n)=\prod_{i=1}^{n-1}(3^i+1)$.

References

[a1] 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 Zbl 0888.57014
[a2] Q. Chen, "The $3$-move conjecture for $5$-braids" , Knots in Hellas '98 (Proc. Internat. Conf. Knot Theory and Its Ramifications , Knots and Everything , 24 (2000) pp. 36–47 Zbl 0973.57003
[a3] J.H. Przytycki, T. Tsukamoto, "The fourth skein module and the Montesinos–Nakanishi conjecture for $3$-algebraic links" J. Knot Th. Ramifications , 10 (2001) pp.959–982 DOI 10.1142/S0218216501001281 Zbl 1001.57029
[a4] 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 Zbl 0968.57004
How to Cite This Entry:
Montesinos-Nakanishi conjecture. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Montesinos-Nakanishi_conjecture&oldid=16341
This article was adapted from an original article by Jozef Przytycki (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article