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| − | 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:
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| − | 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
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| − | 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;
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| | + | A family of tangles (cf. [[Tangle|Tangle]]) defined recursively for any $n$ as follows: |
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| − | 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.
| + | i) $n$-algebraic tangles is the smallest family of $n$-tangles satisfying |
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| − | 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.
| + | 1) any $n$-tangle with $0$ or $1$ crossing is $n$-algebraic; |
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| − | 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.
| + | 2) if $A$ and $B$ are $n$-algebraic tangles, then $r ^ { i } ( A ) * r ^ { j } ( B )$ is $n$-algebraic for any integers $i$, $j$, where $r$ denotes the rotation of a tangle by the angle $\pi /n$ and $*$ denotes (horizontal) composition of tangles. |
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| − | <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" />.
| + | ii) If in condition 2) above, $B$ is restricted to tangles with no more than $k$ crossings, one obtains the family of $( n , k )$-algebraic tangles. |
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| | + | iii) If an $m$-tangle, $T$, is obtained from an $( n , k )$-algebraic tangle (respectively, an $n$-algebraic tangle) by partially closing $2 n - 2 m$ of its endpoints without a crossing, then $T$ is called an $( n , k )$-algebraic $m$-tangle, respectively an $n$-algebraic $m$-tangle. For $m = 0$ one obtains an $( n , k )$-algebraic link, respectively an $n$-algebraic link. |
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| | + | $2$-algebraic tangles were introduced by J.H. Conway (they are often called algebraic tangles in the sense of Conway or arborescent tangles). The $2$-fold branched covering of $S ^ { 3 }$ with a $2$-algebraic link as a branched set is a Waldhausen graph manifold. Thus, not every link is $2$-algebraic. It is an open problem (as of 2001) to find, for a given $n$, a link which is not $n$-algebraic. The smallest $n$ for which a link $L$ is $n$-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 $8 _ { 18 }$ knot is equal to $3$. |
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| | ====References==== | | ====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> | + | <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 $3$-algebraic links" ''J. Knot Th. Ramifications'' (2001)</td></tr></table> |
A family of tangles (cf. Tangle) defined recursively for any $n$ as follows:
i) $n$-algebraic tangles is the smallest family of $n$-tangles satisfying
1) any $n$-tangle with $0$ or $1$ crossing is $n$-algebraic;
2) if $A$ and $B$ are $n$-algebraic tangles, then $r ^ { i } ( A ) * r ^ { j } ( B )$ is $n$-algebraic for any integers $i$, $j$, where $r$ denotes the rotation of a tangle by the angle $\pi /n$ and $*$ denotes (horizontal) composition of tangles.
ii) If in condition 2) above, $B$ is restricted to tangles with no more than $k$ crossings, one obtains the family of $( n , k )$-algebraic tangles.
iii) If an $m$-tangle, $T$, is obtained from an $( n , k )$-algebraic tangle (respectively, an $n$-algebraic tangle) by partially closing $2 n - 2 m$ of its endpoints without a crossing, then $T$ is called an $( n , k )$-algebraic $m$-tangle, respectively an $n$-algebraic $m$-tangle. For $m = 0$ one obtains an $( n , k )$-algebraic link, respectively an $n$-algebraic link.
$2$-algebraic tangles were introduced by J.H. Conway (they are often called algebraic tangles in the sense of Conway or arborescent tangles). The $2$-fold branched covering of $S ^ { 3 }$ with a $2$-algebraic link as a branched set is a Waldhausen graph manifold. Thus, not every link is $2$-algebraic. It is an open problem (as of 2001) to find, for a given $n$, a link which is not $n$-algebraic. The smallest $n$ for which a link $L$ is $n$-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 $8 _ { 18 }$ knot is equal to $3$.
References
| [a1] | J.H. Conway, "An enumeration of knots and links" J. Leech (ed.) , Computational Problems in Abstract Algebra , Pergamon Press (1969) pp. 329–358 |
| [a2] | J.H. Przytycki, T. Tsukamoto, "The fourth skein module and the Montesinos–Nakanishi conjecture for $3$-algebraic links" J. Knot Th. Ramifications (2001) |