L11a263

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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_14,6,15,5 X_18,8,19,7 X_20,18,21,17 X_16,22,17,21 X_2,9,3,10 X_4,14,5,13 X_6,16,7,15 X_8,20,1,19
Gauss code	{1, -8, 2, -9, 4, -10, 5, -11}, {8, -1, 3, -2, 9, -4, 10, -7, 6, -5, 11, -6, 7, -3}

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

Polynomial invariants

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

3

1

-2

18

5

2

3

16

6

3

-3

14

6

5

1

12

6

7

1

10

4

5

-1

8

3

6

3

6

2

4

-2

4

1

4

3

2

1

-1

0

1

Integral Khovanov Homology

(db, data source)

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