L11a361

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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_2,11,3,12 X_4,16,5,15 X_6,22,7,21 X_20,10,21,9 X_18,8,19,7 X_8,20,9,19 X_10,18,1,17
Gauss code	{1, -5, 2, -6, 4, -7, 9, -10, 8, -11}, {5, -1, 3, -2, 6, -4, 11, -9, 10, -8, 7, -3}

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

Polynomial invariants

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

24

1

-1

22

2

20

4

1

-3

18

6

2

4

16

7

5

-2

14

7

5

2

12

6

7

1

10

5

7

-2

8

3

6

3

6

2

5

-3

4

1

4

3

2

1

-1

0

1

Integral Khovanov Homology

(db, data source)

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