# L9a30

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 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a30 at Knotilus! L9a30 is ${\displaystyle 9_{3}^{2}}$ in the Rolfsen table of links.

### Link Presentations

 Planar diagram presentation X8192 X2,9,3,10 X10,3,11,4 X6718 X16,13,17,14 X14,6,15,5 X4,16,5,15 X18,11,7,12 X12,17,13,18 Gauss code {1, -2, 3, -7, 6, -4}, {4, -1, 2, -3, 8, -9, 5, -6, 7, -5, 9, -8}

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {2u^{2}v^{2}-2u^{2}v-2uv^{2}+3uv-2u-2v+2}{uv}}}$ (db) Jones polynomial ${\displaystyle -{\sqrt {q}}+{\frac {1}{\sqrt {q}}}-{\frac {3}{q^{3/2}}}+{\frac {4}{q^{5/2}}}-{\frac {5}{q^{7/2}}}+{\frac {5}{q^{9/2}}}-{\frac {5}{q^{11/2}}}+{\frac {3}{q^{13/2}}}-{\frac {2}{q^{15/2}}}+{\frac {1}{q^{17/2}}}}$ (db) Signature -3 (db) HOMFLY-PT polynomial ${\displaystyle a^{7}\left(-z^{3}\right)-2a^{7}z+a^{5}z^{5}+3a^{5}z^{3}+2a^{5}z+a^{3}z^{5}+3a^{3}z^{3}+2a^{3}z+a^{3}z^{-1}-az^{3}-3az-az^{-1}}$ (db) Kauffman polynomial ${\displaystyle a^{10}z^{4}-2a^{10}z^{2}+2a^{9}z^{5}-4a^{9}z^{3}+a^{9}z+2a^{8}z^{6}-3a^{8}z^{4}+a^{8}z^{2}+2a^{7}z^{7}-5a^{7}z^{5}+7a^{7}z^{3}-2a^{7}z+a^{6}z^{8}-2a^{6}z^{6}+4a^{6}z^{4}-a^{6}z^{2}+3a^{5}z^{7}-9a^{5}z^{5}+12a^{5}z^{3}-4a^{5}z+a^{4}z^{8}-3a^{4}z^{6}+6a^{4}z^{4}-5a^{4}z^{2}+a^{3}z^{7}-a^{3}z^{5}-3a^{3}z^{3}+3a^{3}z-a^{3}z^{-1}+a^{2}z^{6}-2a^{2}z^{4}-a^{2}z^{2}+a^{2}+az^{5}-4az^{3}+4az-az^{-1}}$ (db)

### Khovanov Homology

The coefficients of the monomials ${\displaystyle t^{r}q^{j}}$ are shown, along with their alternating sums ${\displaystyle \chi }$ (fixed ${\displaystyle j}$, alternation over ${\displaystyle r}$).
 \ r \ j \
-7-6-5-4-3-2-1012χ
2         11
0          0
-2       31 2
-4      21  -1
-6     32   1
-8    33    0
-10   22     0
-12  13      2
-14 12       -1
-16 1        1
-181         -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-4}$ ${\displaystyle i=-2}$ ${\displaystyle r=-7}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory. See A Sample KnotTheory Session.

### Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

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