L11a131

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)\left(3v^{2}-2v+3\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$q^{21/2}-3q^{19/2}+5q^{17/2}-7q^{15/2}+9q^{13/2}-10q^{11/2}+10q^{9/2}-8q^{7/2}+5q^{5/2}-4q^{3/2}+{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-9}+za^{-9}-z^{5}a^{-7}-2z^{3}a^{-7}-za^{-7}-z^{5}a^{-5}-z^{3}a^{-5}+za^{-5}+a^{-5}z^{-1}-z^{5}a^{-3}-3z^{3}a^{-3}-4za^{-3}-3a^{-3}z^{-1}+z^{3}a^{-1}+3za^{-1}+2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-6}-z^{10}a^{-8}-z^{9}a^{-5}-4z^{9}a^{-7}-3z^{9}a^{-9}-z^{8}a^{-4}+3z^{8}a^{-6}-4z^{8}a^{-10}-z^{7}a^{-3}+z^{7}a^{-5}+15z^{7}a^{-7}+10z^{7}a^{-9}-3z^{7}a^{-11}-z^{6}a^{-2}-8z^{6}a^{-6}+6z^{6}a^{-8}+14z^{6}a^{-10}-z^{6}a^{-12}-z^{5}a^{-1}-z^{5}a^{-3}-3z^{5}a^{-5}-24z^{5}a^{-7}-11z^{5}a^{-9}+10z^{5}a^{-11}+z^{4}a^{-2}-z^{4}a^{-4}+6z^{4}a^{-6}-7z^{4}a^{-8}-12z^{4}a^{-10}+3z^{4}a^{-12}+4z^{3}a^{-1}+7z^{3}a^{-3}+3z^{3}a^{-5}+13z^{3}a^{-7}+7z^{3}a^{-9}-6z^{3}a^{-11}+3z^{2}a^{-2}+5z^{2}a^{-4}-z^{2}a^{-6}+z^{2}a^{-8}+3z^{2}a^{-10}-z^{2}a^{-12}-5za^{-1}-7za^{-3}-2za^{-5}-2za^{-7}-2za^{-9}-3a^{-2}-3a^{-4}-a^{-6}+2a^{-1}z^{-1}+3a^{-3}z^{-1}+a^{-5}z^{-1}$ (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		\

-2

-1

0

1

2

3

4

5

6

7

8

9

χ

22

1

-1

20

2

18

3

1

-2

16

4

2

14

5

3

-2

12

5

4

1

10

5

0

8

3

5

-2

6

2

5

3

4

2

3

-1

2

1

4

3

0

-2

1

Integral Khovanov Homology

(db, data source)

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