L11a384

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Link Presentations

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{4}v^{2}-u^{4}v+3u^{3}v^{3}-8u^{3}v^{2}+5u^{3}v-u^{3}+u^{2}v^{4}-8u^{2}v^{3}+13u^{2}v^{2}-8u^{2}v+u^{2}-uv^{4}+5uv^{3}-8uv^{2}+3uv-v^{3}+v^{2}}{u^{2}v^{2}}}$ (db)
Jones polynomial	$22q^{9/2}-19q^{7/2}+14q^{5/2}-8q^{3/2}+q^{21/2}-4q^{19/2}+9q^{17/2}-15q^{15/2}+19q^{13/2}-23q^{11/2}+3{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-9}-z^{5}a^{-7}+z^{3}a^{-7}+2za^{-7}-a^{-7}z^{-1}-3z^{5}a^{-5}-6z^{3}a^{-5}-3za^{-5}+a^{-5}z^{-1}-z^{5}a^{-3}+z^{3}a^{-3}+3za^{-3}+z^{3}a^{-1}+za^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-6}-2z^{10}a^{-8}-6z^{9}a^{-5}-12z^{9}a^{-7}-6z^{9}a^{-9}-8z^{8}a^{-4}-13z^{8}a^{-6}-12z^{8}a^{-8}-7z^{8}a^{-10}-6z^{7}a^{-3}+2z^{7}a^{-5}+18z^{7}a^{-7}+6z^{7}a^{-9}-4z^{7}a^{-11}-3z^{6}a^{-2}+13z^{6}a^{-4}+33z^{6}a^{-6}+33z^{6}a^{-8}+15z^{6}a^{-10}-z^{6}a^{-12}-z^{5}a^{-1}+9z^{5}a^{-3}+8z^{5}a^{-5}-4z^{5}a^{-7}+7z^{5}a^{-9}+9z^{5}a^{-11}+4z^{4}a^{-2}-13z^{4}a^{-4}-28z^{4}a^{-6}-21z^{4}a^{-8}-8z^{4}a^{-10}+2z^{4}a^{-12}+2z^{3}a^{-1}-7z^{3}a^{-3}-8z^{3}a^{-5}+2z^{3}a^{-7}-5z^{3}a^{-9}-6z^{3}a^{-11}-z^{2}a^{-2}+6z^{2}a^{-4}+10z^{2}a^{-6}+5z^{2}a^{-8}+z^{2}a^{-10}-z^{2}a^{-12}-za^{-1}+3za^{-3}-za^{-5}-5za^{-7}+za^{-9}+za^{-11}-a^{-6}+a^{-5}z^{-1}+a^{-7}z^{-1}$ (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		\

-2

-1

0

1

2

3

4

5

6

7

8

9

χ

22

1

-1

20

3

18

6

1

-5

16

9

3

6

14

11

7

-4

12

8

4

10

11

1

8

9

12

-3

6

5

10

5

4

3

9

-6

2

1

6

5

0

2

-2

1

Integral Khovanov Homology

(db, data source)

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