L11a363

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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_2,11,3,12 X_4,22,5,21 X_20,10,21,9 X_16,6,17,5 X_8,18,9,17 X_18,8,19,7 X_6,20,7,19 X_10,16,1,15
Gauss code	{1, -4, 2, -5, 7, -10, 9, -8, 6, -11}, {4, -1, 3, -2, 11, -7, 8, -9, 10, -6, 5, -3}

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

Polynomial invariants

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

22

1

-1

20

2

18

4

1

-3

16

7

2

5

14

8

5

-3

12

9

6

3

10

8

0

8

7

9

-2

6

4

8

4

3

7

-4

2

1

5

4

0

2

-2

1

Integral Khovanov Homology

(db, data source)

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