L11a222

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_12,4,13,3 X_22,10,7,9 X_20,12,21,11 X_10,22,11,21 X_16,6,17,5 X_18,16,19,15 X_14,20,15,19 X₂₇₃₈ X_4,14,5,13 X_6,18,1,17
Gauss code	{1, -9, 2, -10, 6, -11}, {9, -1, 3, -5, 4, -2, 10, -8, 7, -6, 11, -7, 8, -4, 5, -3}

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

Polynomial invariants

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

-2

-1

0

1

2

3

4

5

6

7

8

9

χ

22

1

-1

20

2

18

4

1

-3

16

6

2

4

14

7

4

-3

12

8

6

2

10

7

0

8

6

8

-2

6

4

8

4

2

5

-3

2

1

5

4

0

1

-1

-2

1

Integral Khovanov Homology

(db, data source)

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