L11a360

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{4}v^{4}-u^{4}v^{3}-u^{3}v^{4}+u^{3}v^{3}-u^{3}v^{2}-u^{2}v^{3}+u^{2}v^{2}-u^{2}v-uv^{2}+uv-u-v+1}{u^{2}v^{2}}}$ (db)
Jones polynomial	$2q^{9/2}-2q^{7/2}+q^{5/2}-q^{3/2}+q^{25/2}-2q^{23/2}+2q^{21/2}-3q^{19/2}+3q^{17/2}-3q^{15/2}+3q^{13/2}-3q^{11/2}$ (db)
Signature	7 (db)
HOMFLY-PT polynomial	$-z^{9}a^{-7}+z^{7}a^{-5}-8z^{7}a^{-7}+z^{7}a^{-9}+7z^{5}a^{-5}-22z^{5}a^{-7}+6z^{5}a^{-9}+15z^{3}a^{-5}-25z^{3}a^{-7}+10z^{3}a^{-9}+10za^{-5}-11za^{-7}+4za^{-9}+a^{-5}z^{-1}-a^{-7}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-6}-z^{10}a^{-8}-z^{9}a^{-5}-3z^{9}a^{-7}-2z^{9}a^{-9}+7z^{8}a^{-6}+5z^{8}a^{-8}-2z^{8}a^{-10}+8z^{7}a^{-5}+21z^{7}a^{-7}+11z^{7}a^{-9}-2z^{7}a^{-11}-15z^{6}a^{-6}-5z^{6}a^{-8}+8z^{6}a^{-10}-2z^{6}a^{-12}-22z^{5}a^{-5}-48z^{5}a^{-7}-18z^{5}a^{-9}+6z^{5}a^{-11}-2z^{5}a^{-13}+9z^{4}a^{-6}-3z^{4}a^{-8}-6z^{4}a^{-10}+4z^{4}a^{-12}-2z^{4}a^{-14}+25z^{3}a^{-5}+42z^{3}a^{-7}+11z^{3}a^{-9}-2z^{3}a^{-11}+2z^{3}a^{-13}-2z^{3}a^{-15}+2z^{2}a^{-6}+3z^{2}a^{-8}-z^{2}a^{-10}+z^{2}a^{-14}-z^{2}a^{-16}-11za^{-5}-13za^{-7}-3za^{-9}-za^{-11}+za^{-13}+za^{-15}-a^{-6}+a^{-5}z^{-1}+a^{-7}z^{-1}$ (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

26

1

-1

24

1

22

1

0

20

2

1

18

2

0

16

1

0

14

2

0

12

1

0

10

1

2

1

8

1

0

6

1

2

1

4

0

2

1

Integral Khovanov Homology

(db, data source)

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