# L11n79

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 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n79 at Knotilus!

### Link Presentations

 Planar diagram presentation X6172 X16,9,17,10 X4,21,1,22 X11,14,12,15 X3,10,4,11 X5,13,6,12 X13,5,14,22 X15,2,16,3 X20,18,21,17 X18,8,19,7 X8,20,9,19 Gauss code {1, 8, -5, -3}, {-6, -1, 10, -11, 2, 5, -4, 6, -7, 4, -8, -2, 9, -10, 11, -9, 3, 7}

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {uv^{5}-4uv^{4}+6uv^{3}-4uv^{2}-4v^{3}+6v^{2}-4v+1}{{\sqrt {u}}v^{5/2}}}}$ (db) Jones polynomial ${\displaystyle q^{9/2}-{\frac {2}{q^{9/2}}}-3q^{7/2}+{\frac {4}{q^{7/2}}}+6q^{5/2}-{\frac {7}{q^{5/2}}}-8q^{3/2}+{\frac {9}{q^{3/2}}}+9{\sqrt {q}}-{\frac {11}{\sqrt {q}}}}$ (db) Signature -1 (db) HOMFLY-PT polynomial ${\displaystyle a^{5}z+a^{5}z^{-1}-a^{3}z^{5}-4a^{3}z^{3}+z^{3}a^{-3}-5a^{3}z+2za^{-3}-2a^{3}z^{-1}+a^{-3}z^{-1}+az^{7}+5az^{5}-2z^{5}a^{-1}+9az^{3}-7z^{3}a^{-1}+7az-7za^{-1}+3az^{-1}-3a^{-1}z^{-1}}$ (db) Kauffman polynomial ${\displaystyle -az^{9}-z^{9}a^{-1}-3a^{2}z^{8}-3z^{8}a^{-2}-6z^{8}-3a^{3}z^{7}-6az^{7}-6z^{7}a^{-1}-3z^{7}a^{-3}-a^{4}z^{6}+6a^{2}z^{6}+5z^{6}a^{-2}-z^{6}a^{-4}+13z^{6}+7a^{3}z^{5}+24az^{5}+26z^{5}a^{-1}+9z^{5}a^{-3}-2a^{4}z^{4}-10a^{2}z^{4}+6z^{4}a^{-2}+3z^{4}a^{-4}-5z^{4}-3a^{5}z^{3}-15a^{3}z^{3}-32az^{3}-27z^{3}a^{-1}-7z^{3}a^{-3}+2a^{4}z^{2}+7a^{2}z^{2}-8z^{2}a^{-2}-3z^{2}a^{-4}+4a^{5}z+10a^{3}z+16az+13za^{-1}+3za^{-3}-2a^{2}+2a^{-2}+a^{-4}-a^{5}z^{-1}-2a^{3}z^{-1}-3az^{-1}-3a^{-1}z^{-1}-a^{-3}z^{-1}}$ (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 \
-4-3-2-1012345χ
10         1-1
8        2 2
6       41 -3
4      42  2
2     54   -1
0    64    2
-2   46     2
-4  35      -2
-6 14       3
-813        -2
-102         2
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-2}$ ${\displaystyle i=0}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### 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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