L11a299

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_12,4,13,3 X_22,12,9,11 X_16,6,17,5 X_20,16,21,15 X_14,22,15,21 X_18,8,19,7 X_2,9,3,10 X_4,14,5,13 X_6,18,7,17 X_8,20,1,19
Gauss code	{1, -8, 2, -9, 4, -10, 7, -11}, {8, -1, 3, -2, 9, -6, 5, -4, 10, -7, 11, -5, 6, -3}

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

Polynomial invariants

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

24

1

-1

22

2

20

4

1

-3

18

6

2

4

16

6

4

-2

14

7

6

1

12

7

0

10

4

6

-2

8

3

7

4

6

2

4

-2

4

1

4

3

2

1

-1

0

1

Integral Khovanov Homology

(db, data source)

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