# L11a524

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 Planar diagram presentation X8192 X22,10,13,9 X18,8,19,7 X20,17,21,18 X6,22,1,21 X4,11,5,12 X16,6,17,5 X10,16,11,15 X12,20,7,19 X2,13,3,14 X14,3,15,4 Gauss code {1, -10, 11, -6, 7, -5}, {3, -1, 2, -8, 6, -9}, {10, -11, 8, -7, 4, -3, 9, -4, 5, -2}

### Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {(v-1)\left(u^{2}vw^{3}-3u^{2}vw^{2}+2u^{2}vw-u^{2}w^{3}+2u^{2}w^{2}-u^{2}w-uvw^{3}+4uvw^{2}-5uvw+uv+uw^{3}-5uw^{2}+4uw-u-vw^{2}+2vw-v+2w^{2}-3w+1\right)}{uvw^{3/2}}}}$ (db) Jones polynomial ${\displaystyle q^{6}-4q^{5}-q^{-5}+10q^{4}+5q^{-4}-16q^{3}-11q^{-3}+24q^{2}+18q^{-2}-26q-24q^{-1}+28}$ (db) Signature 0 (db) HOMFLY-PT polynomial ${\displaystyle z^{4}a^{-4}+2z^{2}a^{-4}+a^{-4}z^{-2}+a^{-4}-a^{2}z^{6}-2z^{6}a^{-2}-2a^{2}z^{4}-6z^{4}a^{-2}-5z^{2}a^{-2}-2a^{-2}z^{-2}+a^{2}-2a^{-2}+z^{8}+4z^{6}+5z^{4}+z^{2}+z^{-2}}$ (db) Kauffman polynomial ${\displaystyle z^{6}a^{-6}-2z^{4}a^{-6}+z^{2}a^{-6}+4z^{7}a^{-5}+a^{5}z^{5}-8z^{5}a^{-5}+4z^{3}a^{-5}+8z^{8}a^{-4}+5a^{4}z^{6}-18z^{6}a^{-4}-4a^{4}z^{4}+16z^{4}a^{-4}-10z^{2}a^{-4}-a^{-4}z^{-2}+4a^{-4}+8z^{9}a^{-3}+11a^{3}z^{7}-11z^{7}a^{-3}-14a^{3}z^{5}-z^{5}a^{-3}+4a^{3}z^{3}+6z^{3}a^{-3}-5za^{-3}+2a^{-3}z^{-1}+3z^{10}a^{-2}+14a^{2}z^{8}+16z^{8}a^{-2}-19a^{2}z^{6}-52z^{6}a^{-2}+6a^{2}z^{4}+50z^{4}a^{-2}-22z^{2}a^{-2}-2a^{-2}z^{-2}-a^{2}+6a^{-2}+10az^{9}+18z^{9}a^{-1}-2az^{7}-28z^{7}a^{-1}-20az^{5}+2z^{5}a^{-1}+13az^{3}+11z^{3}a^{-1}-5za^{-1}+2a^{-1}z^{-1}+3z^{10}+22z^{8}-57z^{6}+42z^{4}-11z^{2}-z^{-2}+2}$ (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 \
-5-4-3-2-10123456χ
13           11
11          3 -3
9         71 6
7        104  -6
5       146   8
3      1311    -2
1     1513     2
-1    1014      4
-3   814       -6
-5  411        7
-7 17         -6
-9 4          4
-111           -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-1}$ ${\displaystyle i=1}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}}$ ${\displaystyle {\mathbb {Z} }^{8}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{10}}$ ${\displaystyle {\mathbb {Z} }^{10}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{14}}$ ${\displaystyle {\mathbb {Z} }^{15}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{13}}$ ${\displaystyle {\mathbb {Z} }^{13}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{13}}$ ${\displaystyle {\mathbb {Z} }^{14}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}}$ ${\displaystyle {\mathbb {Z} }^{10}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}}$ ${\displaystyle {\mathbb {Z} }^{7}}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=6}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### Computer Talk

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