# L11n94

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 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n94 at Knotilus!

### Link Presentations

 Planar diagram presentation X6172 X12,4,13,3 X9,14,10,15 X19,22,20,5 X11,21,12,20 X21,11,22,10 X15,19,16,18 X7,17,8,16 X17,9,18,8 X2536 X4,14,1,13 Gauss code {1, -10, 2, -11}, {10, -1, -8, 9, -3, 6, -5, -2, 11, 3, -7, 8, -9, 7, -4, 5, -6, 4}

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle -{\frac {(u-1)(v-1)\left(v^{2}+1\right)}{{\sqrt {u}}v^{3/2}}}}$ (db) Jones polynomial ${\displaystyle q^{9/2}-2q^{7/2}+2q^{5/2}-{\frac {1}{q^{5/2}}}-4q^{3/2}+{\frac {2}{q^{3/2}}}+q^{15/2}-q^{13/2}+3{\sqrt {q}}-{\frac {3}{\sqrt {q}}}}$ (db) Signature 3 (db) HOMFLY-PT polynomial ${\displaystyle za^{-7}+a^{-7}z^{-1}-z^{3}a^{-5}-4za^{-5}-3a^{-5}z^{-1}+2z^{3}a^{-3}+6za^{-3}+3a^{-3}z^{-1}-z^{5}a^{-1}+az^{3}-4z^{3}a^{-1}+2az-5za^{-1}+az^{-1}-2a^{-1}z^{-1}}$ (db) Kauffman polynomial ${\displaystyle -z^{9}a^{-1}-z^{9}a^{-3}-4z^{8}a^{-2}-2z^{8}a^{-4}-2z^{8}-az^{7}+2z^{7}a^{-1}+3z^{7}a^{-3}-z^{7}a^{-5}-z^{7}a^{-7}+21z^{6}a^{-2}+12z^{6}a^{-4}-z^{6}a^{-8}+10z^{6}+5az^{5}+9z^{5}a^{-1}+6z^{5}a^{-3}+8z^{5}a^{-5}+6z^{5}a^{-7}-29z^{4}a^{-2}-18z^{4}a^{-4}+3z^{4}a^{-6}+5z^{4}a^{-8}-13z^{4}-7az^{3}-19z^{3}a^{-1}-18z^{3}a^{-3}-14z^{3}a^{-5}-8z^{3}a^{-7}+13z^{2}a^{-2}+8z^{2}a^{-4}-5z^{2}a^{-6}-5z^{2}a^{-8}+5z^{2}+4az+11za^{-1}+14za^{-3}+10za^{-5}+3za^{-7}-2a^{-2}+2a^{-6}+a^{-8}-az^{-1}-2a^{-1}z^{-1}-3a^{-3}z^{-1}-3a^{-5}z^{-1}-a^{-7}z^{-1}}$ (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 \
-4-3-2-101234567χ
16           1-1
14            0
12        111 1
10       21   -1
8      221   1
6     231    0
4    321     2
2   241      1
0  121       0
-2 12         1
-4 1          -1
-61           1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=0}$ ${\displaystyle i=2}$ ${\displaystyle i=4}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=6}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=7}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### 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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