L11a305

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Link Presentations

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

A Braid Representative

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Polynomial invariants

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

-3

-2

-1

0

1

2

3

4

5

6

7

8

χ

20

1

18

2

-2

16

4

1

3

14

6

2

-4

12

7

4

3

10

8

7

-1

8

7

6

1

6

5

8

3

4

7

-3

2

6

4

0

1

3

-2

2

-4

1

-1

Integral Khovanov Homology

(db, data source)

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