L11a68

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)\left(4v^{2}-5v+4\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$q^{21/2}-3q^{19/2}+7q^{17/2}-11q^{15/2}+15q^{13/2}-17q^{11/2}+16q^{9/2}-15q^{7/2}+10q^{5/2}-6q^{3/2}+2{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-9}+za^{-9}+a^{-9}z^{-1}-z^{5}a^{-7}-z^{3}a^{-7}-2za^{-7}-2a^{-7}z^{-1}-2z^{5}a^{-5}-3z^{3}a^{-5}+a^{-5}z^{-1}-z^{5}a^{-3}-z^{3}a^{-3}-za^{-3}-a^{-3}z^{-1}+z^{3}a^{-1}+2za^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-12}-3z^{4}a^{-12}+2z^{2}a^{-12}+3z^{7}a^{-11}-8z^{5}a^{-11}+5z^{3}a^{-11}+5z^{8}a^{-10}-14z^{6}a^{-10}+14z^{4}a^{-10}-9z^{2}a^{-10}+3a^{-10}+4z^{9}a^{-9}-7z^{7}a^{-9}+4z^{5}a^{-9}-6z^{3}a^{-9}+3za^{-9}-a^{-9}z^{-1}+z^{10}a^{-8}+9z^{8}a^{-8}-30z^{6}a^{-8}+37z^{4}a^{-8}-26z^{2}a^{-8}+7a^{-8}+7z^{9}a^{-7}-13z^{7}a^{-7}+12z^{5}a^{-7}-10z^{3}a^{-7}+6za^{-7}-2a^{-7}z^{-1}+z^{10}a^{-6}+8z^{8}a^{-6}-22z^{6}a^{-6}+28z^{4}a^{-6}-17z^{2}a^{-6}+4a^{-6}+3z^{9}a^{-5}-3z^{5}a^{-5}+z^{3}a^{-5}+3za^{-5}-a^{-5}z^{-1}+4z^{8}a^{-4}-5z^{6}a^{-4}+5z^{4}a^{-4}-2z^{2}a^{-4}+3z^{7}a^{-3}-2z^{5}a^{-3}-3z^{3}a^{-3}+3za^{-3}-a^{-3}z^{-1}+2z^{6}a^{-2}-3z^{4}a^{-2}+a^{-2}+z^{5}a^{-1}-3z^{3}a^{-1}+3za^{-1}-a^{-1}z^{-1}$ (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

5

1

-4

16

6

2

4

14

9

5

-4

12

8

6

2

10

8

9

1

8

7

8

-1

6

3

8

5

4

3

7

-4

2

1

5

4

0

1

-1

-2

1

Integral Khovanov Homology

(db, data source)

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