L11a242

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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_14,12,15,11 X_10,16,11,15 X_18,6,19,5 X_20,18,21,17 X_16,22,17,21 X₂₇₃₈ X_4,14,5,13 X_6,20,1,19
Gauss code	{1, -9, 2, -10, 6, -11}, {9, -1, 3, -5, 4, -2, 10, -4, 5, -8, 7, -6, 11, -7, 8, -3}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {4t(2)^{2}t(1)^{2}-7t(2)t(1)^{2}+3t(1)^{2}-7t(2)^{2}t(1)+13t(2)t(1)-7t(1)+3t(2)^{2}-7t(2)+4}{t(1)t(2)}}$ (db)
Jones polynomial	$q^{21/2}-4q^{19/2}+8q^{17/2}-12q^{15/2}+16q^{13/2}-18q^{11/2}+17q^{9/2}-15q^{7/2}+10q^{5/2}-6q^{3/2}+2{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-9}-z^{5}a^{-7}+za^{-7}-2z^{5}a^{-5}-3z^{3}a^{-5}-za^{-5}-z^{5}a^{-3}-z^{3}a^{-3}-za^{-3}-a^{-3}z^{-1}+z^{3}a^{-1}+2za^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-6}-z^{10}a^{-8}-3z^{9}a^{-5}-7z^{9}a^{-7}-4z^{9}a^{-9}-4z^{8}a^{-4}-9z^{8}a^{-6}-11z^{8}a^{-8}-6z^{8}a^{-10}-3z^{7}a^{-3}-z^{7}a^{-5}+7z^{7}a^{-7}+z^{7}a^{-9}-4z^{7}a^{-11}-2z^{6}a^{-2}+5z^{6}a^{-4}+23z^{6}a^{-6}+31z^{6}a^{-8}+14z^{6}a^{-10}-z^{6}a^{-12}-z^{5}a^{-1}+2z^{5}a^{-3}+7z^{5}a^{-5}+10z^{5}a^{-7}+16z^{5}a^{-9}+10z^{5}a^{-11}+3z^{4}a^{-2}-6z^{4}a^{-4}-23z^{4}a^{-6}-23z^{4}a^{-8}-7z^{4}a^{-10}+2z^{4}a^{-12}+3z^{3}a^{-1}+3z^{3}a^{-3}-9z^{3}a^{-5}-16z^{3}a^{-7}-13z^{3}a^{-9}-6z^{3}a^{-11}+4z^{2}a^{-4}+7z^{2}a^{-6}+5z^{2}a^{-8}+z^{2}a^{-10}-z^{2}a^{-12}-3za^{-1}-3za^{-3}+3za^{-5}+5za^{-7}+3za^{-9}+za^{-11}-a^{-2}+a^{-1}z^{-1}+a^{-3}z^{-1}$ (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

3

18

5

1

-4

16

7

3

4

14

9

5

-4

12

9

7

2

10

8

9

1

8

7

9

-2

6

4

9

5

4

2

6

-4

2

1

5

4

0

1

-1

-2

1

Integral Khovanov Homology

(db, data source)

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