L11a22

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,4,11,3 X_12,8,13,7 X_18,10,19,9 X_22,14,5,13 X_20,16,21,15 X_16,20,17,19 X_14,22,15,21 X_8,18,9,17 X₂₅₃₆ X_4,12,1,11
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 4, -2, 11, -3, 5, -8, 6, -7, 9, -4, 7, -6, 8, -5}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(3t(2)^{2}-4t(2)+3\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$12q^{9/2}-11q^{7/2}+7q^{5/2}-5q^{3/2}+q^{21/2}-3q^{19/2}+6q^{17/2}-8q^{15/2}+11q^{13/2}-13q^{11/2}+2{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-3}-z^{5}a^{-5}-z^{5}a^{-7}+z^{3}a^{-1}-2z^{3}a^{-3}-2z^{3}a^{-7}+z^{3}a^{-9}+2za^{-1}-2za^{-3}+2za^{-5}-3za^{-7}+za^{-9}+a^{-1}z^{-1}-a^{-3}z^{-1}+a^{-5}z^{-1}-2a^{-7}z^{-1}+a^{-9}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-6}-z^{10}a^{-8}-2z^{9}a^{-5}-6z^{9}a^{-7}-4z^{9}a^{-9}-2z^{8}a^{-4}-2z^{8}a^{-6}-5z^{8}a^{-8}-5z^{8}a^{-10}-2z^{7}a^{-3}+3z^{7}a^{-5}+21z^{7}a^{-7}+13z^{7}a^{-9}-3z^{7}a^{-11}-2z^{6}a^{-2}+11z^{6}a^{-6}+28z^{6}a^{-8}+18z^{6}a^{-10}-z^{6}a^{-12}-z^{5}a^{-1}-8z^{5}a^{-5}-32z^{5}a^{-7}-14z^{5}a^{-9}+9z^{5}a^{-11}+4z^{4}a^{-2}+3z^{4}a^{-4}-23z^{4}a^{-6}-44z^{4}a^{-8}-19z^{4}a^{-10}+3z^{4}a^{-12}+3z^{3}a^{-1}+7z^{3}a^{-3}+12z^{3}a^{-5}+20z^{3}a^{-7}+8z^{3}a^{-9}-4z^{3}a^{-11}-z^{2}a^{-2}+17z^{2}a^{-6}+27z^{2}a^{-8}+10z^{2}a^{-10}-z^{2}a^{-12}-3za^{-1}-5za^{-3}-7za^{-5}-8za^{-7}-3za^{-9}-a^{-2}-4a^{-6}-7a^{-8}-3a^{-10}+a^{-1}z^{-1}+a^{-3}z^{-1}+a^{-5}z^{-1}+2a^{-7}z^{-1}+a^{-9}z^{-1}$ (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

4

2

14

7

4

-3

12

6

4

2

10

6

7

1

8

5

6

-1

6

2

6

4

3

5

-2

2

1

4

3

0

1

-1

-2

1

Integral Khovanov Homology

(db, data source)

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