# L11a290

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 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a290 at Knotilus!

### Link Presentations

 Planar diagram presentation X10,1,11,2 X20,13,21,14 X12,4,13,3 X2,19,3,20 X14,7,15,8 X16,5,17,6 X6,15,7,16 X8,9,1,10 X18,12,19,11 X22,18,9,17 X4,22,5,21 Gauss code {1, -4, 3, -11, 6, -7, 5, -8}, {8, -1, 9, -3, 2, -5, 7, -6, 10, -9, 4, -2, 11, -10}

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {u^{3}v^{4}-3u^{3}v^{3}+3u^{3}v^{2}-u^{3}v+u^{2}v^{5}-4u^{2}v^{4}+8u^{2}v^{3}-9u^{2}v^{2}+4u^{2}v-u^{2}-uv^{5}+4uv^{4}-9uv^{3}+8uv^{2}-4uv+u-v^{4}+3v^{3}-3v^{2}+v}{u^{3/2}v^{5/2}}}}$ (db) Jones polynomial ${\displaystyle -q^{7/2}+4q^{5/2}-9q^{3/2}+15{\sqrt {q}}-{\frac {20}{\sqrt {q}}}+{\frac {22}{q^{3/2}}}-{\frac {23}{q^{5/2}}}+{\frac {19}{q^{7/2}}}-{\frac {14}{q^{9/2}}}+{\frac {8}{q^{11/2}}}-{\frac {4}{q^{13/2}}}+{\frac {1}{q^{15/2}}}}$ (db) Signature -1 (db) HOMFLY-PT polynomial ${\displaystyle a^{3}z^{7}+az^{7}-a^{5}z^{5}+3a^{3}z^{5}+3az^{5}-z^{5}a^{-1}-2a^{5}z^{3}+2a^{3}z^{3}+3az^{3}-2z^{3}a^{-1}-2a^{3}z+az-za^{-1}+a^{5}z^{-1}-a^{3}z^{-1}}$ (db) Kauffman polynomial ${\displaystyle a^{8}z^{6}-2a^{8}z^{4}+4a^{7}z^{7}-10a^{7}z^{5}+5a^{7}z^{3}+7a^{6}z^{8}-18a^{6}z^{6}+13a^{6}z^{4}-3a^{6}z^{2}+7a^{5}z^{9}-15a^{5}z^{7}+9a^{5}z^{5}-a^{5}z^{3}-2a^{5}z+a^{5}z^{-1}+3a^{4}z^{10}+6a^{4}z^{8}-29a^{4}z^{6}+30a^{4}z^{4}-8a^{4}z^{2}-a^{4}+15a^{3}z^{9}-37a^{3}z^{7}+36a^{3}z^{5}+z^{5}a^{-3}-13a^{3}z^{3}-z^{3}a^{-3}-a^{3}z+a^{3}z^{-1}+3a^{2}z^{10}+9a^{2}z^{8}-30a^{2}z^{6}+4z^{6}a^{-2}+29a^{2}z^{4}-5z^{4}a^{-2}-9a^{2}z^{2}+z^{2}a^{-2}+8az^{9}-10az^{7}+8z^{7}a^{-1}+4az^{5}-12z^{5}a^{-1}-az^{3}+5z^{3}a^{-1}-za^{-1}+10z^{8}-16z^{6}+9z^{4}-3z^{2}}$ (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 \
-7-6-5-4-3-2-101234χ
8           11
6          3 -3
4         61 5
2        93  -6
0       116   5
-2      1210    -2
-4     1110     1
-6    812      4
-8   611       -5
-10  39        6
-12 15         -4
-14 3          3
-161           -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-2}$ ${\displaystyle i=0}$ ${\displaystyle r=-7}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}}$ ${\displaystyle {\mathbb {Z} }^{8}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}}$ ${\displaystyle {\mathbb {Z} }^{11}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}}$ ${\displaystyle {\mathbb {Z} }^{12}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}}$ ${\displaystyle {\mathbb {Z} }^{11}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}}$ ${\displaystyle {\mathbb {Z} }^{9}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### 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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