Difference between revisions of "Montesinos-Nakanishi conjecture"
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| − | Any [[Link|link]] can be reduced to a trivial link by a sequence of | + | {{TEX|done}} |
| + | Any [[Link|link]] can be reduced to a trivial link by a sequence of $3$-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 | + | 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 | + | 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> | <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> | ||
Revision as of 11:51, 29 June 2014
Any link can be reduced to a trivial link by a sequence of $3$-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 $12$ 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 |
| [a2] | Q. Chen, "The  -move conjecture for  -braids" , Knots in Hellas '98 (Proc. Internat. Conf. Knot Theory and Its Ramifications , Knots and Everything , 24 (2000) pp. 36–47 |
| [a3] | J.H. Przytycki, T. Tsukamoto, "The fourth skein module and the Montesinos–Nakanishi conjecture for  -algebraic links" J. Knot Th. Ramifications , to appear (2001) |
| [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 |
How to Cite This Entry:
Montesinos-Nakanishi conjecture. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Montesinos-Nakanishi_conjecture&oldid=22823
Montesinos-Nakanishi conjecture. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Montesinos-Nakanishi_conjecture&oldid=22823
This article was adapted from an original article by Jozef Przytycki (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article