View source for Algebraic tangles

A family of tangles (cf. [[Tangle|Tangle]]) defined recursively for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a1301901.png" /> as follows:

i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a1301903.png" />-algebraic tangles is the smallest family of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a1301904.png" />-tangles satisfying

1) any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a1301905.png" />-tangle with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a1301906.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a1301907.png" /> crossing is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a1301908.png" />-algebraic;

2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a1301909.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a13019010.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a13019011.png" />-algebraic tangles, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a13019012.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a13019013.png" />-algebraic for any integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a13019014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a13019015.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a13019016.png" /> denotes the rotation of a tangle by the angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a13019017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a13019018.png" /> denotes (horizontal) composition of tangles.

ii) If in condition 2) above, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a13019019.png" /> is restricted to tangles with no more than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a13019020.png" /> crossings, one obtains the family of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a13019022.png" />-algebraic tangles.

iii) If an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a13019023.png" />-tangle, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a13019024.png" />, is obtained from an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a13019025.png" />-algebraic tangle (respectively, an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a13019026.png" />-algebraic tangle) by partially closing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a13019027.png" /> of its endpoints without a crossing, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a13019028.png" /> is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a13019031.png" />-algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a13019032.png" />-tangle, respectively an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a13019035.png" />-algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a13019036.png" />-tangle. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a13019037.png" /> one obtains an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a13019039.png" />-algebraic link, respectively an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a13019041.png" />-algebraic link.

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a13019042.png" />-algebraic tangles were introduced by J.H. Conway (they are often called algebraic tangles in the sense of Conway or arborescent tangles). The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a13019043.png" />-fold branched covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a13019044.png" /> with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a13019045.png" />-algebraic link as a branched set is a Waldhausen graph manifold. Thus, not every link is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a13019046.png" />-algebraic. It is an open problem (as of 2001) to find, for a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a13019047.png" />, a link which is not <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a13019048.png" />-algebraic. The smallest <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a13019049.png" /> for which a link <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a13019050.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a13019051.png" />-algebraic is called the algebraic index of the link (it is bounded from above by the braid and bridge indices of the link). For example, the algebraic index of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a13019052.png" /> knot is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130190/a13019053.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.H. Conway,   "An enumeration of knots and links"  J. Leech (ed.) , ''Computational Problems in Abstract Algebra'' , Pergamon Press  (1969)  pp. 329–358</TD></TR><TR><TD valign="top">[a2]</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/a/a130/a130190/a13019054.png" />-algebraic links"  ''J. Knot Th. Ramifications''  (2001)</TD></TR></table>