# L11n143

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 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n143 at Knotilus!

### Link Presentations

 Planar diagram presentation X8192 X11,19,12,18 X3,10,4,11 X17,3,18,2 X12,5,13,6 X6718 X16,10,17,9 X20,16,21,15 X22,14,7,13 X14,22,15,21 X4,20,5,19 Gauss code {1, 4, -3, -11, 5, -6}, {6, -1, 7, 3, -2, -5, 9, -10, 8, -7, -4, 2, 11, -8, 10, -9}

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle -{\frac {2u^{2}v^{2}-2u^{2}v+uv-2v+2}{uv}}}$ (db) Jones polynomial ${\displaystyle q^{13/2}-q^{11/2}+2q^{9/2}-3q^{7/2}+2q^{5/2}-3q^{3/2}+2{\sqrt {q}}-{\frac {2}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}-{\frac {1}{q^{5/2}}}}$ (db) Signature -1 (db) HOMFLY-PT polynomial ${\displaystyle z^{3}a^{-5}+3za^{-5}+a^{-5}z^{-1}-z^{5}a^{-3}-4z^{3}a^{-3}-4za^{-3}-2a^{-3}z^{-1}-z^{5}a^{-1}+az^{3}-4z^{3}a^{-1}+3az-3za^{-1}+az^{-1}}$ (db) Kauffman polynomial ${\displaystyle z^{8}a^{-6}-7z^{6}a^{-6}+16z^{4}a^{-6}-13z^{2}a^{-6}+2a^{-6}+z^{9}a^{-5}-6z^{7}a^{-5}+11z^{5}a^{-5}-8z^{3}a^{-5}+4za^{-5}-a^{-5}z^{-1}+3z^{8}a^{-4}-18z^{6}a^{-4}+33z^{4}a^{-4}-22z^{2}a^{-4}+5a^{-4}+z^{9}a^{-3}-4z^{7}a^{-3}+2z^{5}a^{-3}+a^{3}z+5za^{-3}-2a^{-3}z^{-1}+2z^{8}a^{-2}-10z^{6}a^{-2}+14z^{4}a^{-2}+a^{2}z^{2}-9z^{2}a^{-2}+3a^{-2}+2z^{7}a^{-1}-9z^{5}a^{-1}+2az^{3}+10z^{3}a^{-1}-4az-4za^{-1}+az^{-1}+z^{6}-3z^{4}+z^{2}-1}$ (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 \
-2-101234567χ
14         1-1
12          0
10       21 -1
8      1   1
6     12   1
4    21    1
2    1     1
0  22      0
-2  1       1
-411        0
-61         1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-2}$ ${\displaystyle i=0}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=6}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=7}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### 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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