L11a314

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Planar diagram presentation	X_10,1,11,2 X_20,11,21,12 X_8,21,1,22 X_16,10,17,9 X_14,8,15,7 X_12,4,13,3 X_18,6,19,5 X_6,18,7,17 X_4,14,5,13 X_22,16,9,15 X_2,20,3,19
Gauss code	{1, -11, 6, -9, 7, -8, 5, -3}, {4, -1, 2, -6, 9, -5, 10, -4, 8, -7, 11, -2, 3, -10}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{3}v^{4}-2u^{3}v^{3}+2u^{3}v^{2}-u^{3}v+u^{2}v^{5}-5u^{2}v^{4}+11u^{2}v^{3}-10u^{2}v^{2}+5u^{2}v-u^{2}-uv^{5}+5uv^{4}-10uv^{3}+11uv^{2}-5uv+u-v^{4}+2v^{3}-2v^{2}+v}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$25q^{9/2}-26q^{7/2}+21q^{5/2}-16q^{3/2}+{\frac {1}{q^{3/2}}}-q^{19/2}+4q^{17/2}-10q^{15/2}+17q^{13/2}-22q^{11/2}+9{\sqrt {q}}-{\frac {4}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-3}+z^{7}a^{-5}-z^{5}a^{-1}+3z^{5}a^{-3}+3z^{5}a^{-5}-z^{5}a^{-7}-2z^{3}a^{-1}+4z^{3}a^{-3}+4z^{3}a^{-5}-2z^{3}a^{-7}-za^{-1}+3za^{-3}+2za^{-5}-2za^{-7}+a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$-3z^{10}a^{-4}-3z^{10}a^{-6}-7z^{9}a^{-3}-16z^{9}a^{-5}-9z^{9}a^{-7}-7z^{8}a^{-2}-10z^{8}a^{-4}-15z^{8}a^{-6}-12z^{8}a^{-8}-4z^{7}a^{-1}+9z^{7}a^{-3}+28z^{7}a^{-5}+6z^{7}a^{-7}-9z^{7}a^{-9}+14z^{6}a^{-2}+30z^{6}a^{-4}+38z^{6}a^{-6}+19z^{6}a^{-8}-4z^{6}a^{-10}-z^{6}+9z^{5}a^{-1}+2z^{5}a^{-3}-14z^{5}a^{-5}+7z^{5}a^{-7}+13z^{5}a^{-9}-z^{5}a^{-11}-7z^{4}a^{-2}-19z^{4}a^{-4}-27z^{4}a^{-6}-13z^{4}a^{-8}+4z^{4}a^{-10}+2z^{4}-6z^{3}a^{-1}-4z^{3}a^{-3}+5z^{3}a^{-5}-6z^{3}a^{-7}-8z^{3}a^{-9}+z^{3}a^{-11}+3z^{2}a^{-4}+8z^{2}a^{-6}+5z^{2}a^{-8}-z^{2}a^{-10}-z^{2}+za^{-1}+2za^{-3}-za^{-5}+2za^{-9}+a^{-4}-a^{-3}z^{-1}-a^{-5}z^{-1}$ (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		\

-3

-2

-1

0

1

2

3

4

5

6

7

8

χ

20

1

18

3

-3

16

7

1

6

14

10

3

-7

12

7

5

10

14

11

-3

8

12

11

1

6

9

14

5

4

7

12

-5

2

3

10

7

0

1

6

-5

-2

3

-4

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=2$	$i=4$
$r=-3$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=0$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{7}$
$r=1$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=2$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=3$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{14}$	${\mathbb {Z} }^{14}$
$r=4$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{12}$
$r=5$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=6$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=7$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=8$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$