L11a181

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Planar diagram presentation	X₈₁₉₂ X_20,9,21,10 X_14,5,15,6 X_16,8,17,7 X_10,4,11,3 X_22,14,7,13 X_18,12,19,11 X_12,18,13,17 X_6,15,1,16 X_4,21,5,22 X_2,20,3,19
Gauss code	{1, -11, 5, -10, 3, -9}, {4, -1, 2, -5, 7, -8, 6, -3, 9, -4, 8, -7, 11, -2, 10, -6}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2u^{2}v^{4}-7u^{2}v^{3}+8u^{2}v^{2}-3u^{2}v-2uv^{4}+9uv^{3}-15uv^{2}+9uv-2u-3v^{3}+8v^{2}-7v+2}{uv^{2}}}$ (db)
Jones polynomial	$-q^{11/2}+5q^{9/2}-10q^{7/2}+16q^{5/2}-22q^{3/2}+24{\sqrt {q}}-{\frac {25}{\sqrt {q}}}+{\frac {21}{q^{3/2}}}-{\frac {16}{q^{5/2}}}+{\frac {9}{q^{7/2}}}-{\frac {4}{q^{9/2}}}+{\frac {1}{q^{11/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}+z^{7}a^{-1}-a^{3}z^{5}+3az^{5}+2z^{5}a^{-1}-z^{5}a^{-3}-2a^{3}z^{3}+4az^{3}-2z^{3}a^{-1}-z^{3}a^{-3}-a^{3}z+4az-6za^{-1}+2za^{-3}+2az^{-1}-3a^{-1}z^{-1}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-4z^{10}a^{-2}-4z^{10}-12az^{9}-20z^{9}a^{-1}-8z^{9}a^{-3}-17a^{2}z^{8}-4z^{8}a^{-2}-5z^{8}a^{-4}-16z^{8}-15a^{3}z^{7}+12az^{7}+52z^{7}a^{-1}+24z^{7}a^{-3}-z^{7}a^{-5}-9a^{4}z^{6}+29a^{2}z^{6}+39z^{6}a^{-2}+15z^{6}a^{-4}+62z^{6}-4a^{5}z^{5}+21a^{3}z^{5}+18az^{5}-28z^{5}a^{-1}-19z^{5}a^{-3}+2z^{5}a^{-5}-a^{6}z^{4}+6a^{4}z^{4}-15a^{2}z^{4}-36z^{4}a^{-2}-12z^{4}a^{-4}-46z^{4}+a^{5}z^{3}-12a^{3}z^{3}-22az^{3}-7z^{3}a^{-1}+z^{3}a^{-3}-z^{3}a^{-5}-a^{4}z^{2}+2a^{2}z^{2}+3z^{2}a^{-2}+6z^{2}+2a^{3}z+9az+10za^{-1}+3za^{-3}+3a^{-2}+a^{-4}+3-2az^{-1}-3a^{-1}z^{-1}-a^{-3}z^{-1}$ (db)

Khovanov Homology

The coefficients of the monomials

t^{r}q^{j}

are shown, along with their alternating sums

\chi

(fixed

j

, alternation over

r

).

\		r
	\
j		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

12

1

10

4

-4

8

6

1

5

6

10

4

-6

4

12

6

2

12

10

-2

0

13

12

1

-2

9

13

4

-4

7

12

-5

-6

3

10

7

-8

1

6

-5

-10

3

-12

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{7}$
$r=-1$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=0$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{13}$
$r=1$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=2$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=4$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$