L10a91

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)\left(u^{2}v-u^{2}+uv^{2}-4uv+u-v^{2}+v\right)}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-q^{13/2}+4q^{11/2}-7q^{9/2}+10q^{7/2}-13q^{5/2}+13q^{3/2}-13{\sqrt {q}}+{\frac {9}{\sqrt {q}}}-{\frac {6}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {1}{q^{7/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{-5}+z^{5}a^{-3}+z^{3}a^{-3}+a^{3}z+za^{-3}+z^{5}a^{-1}-2az^{3}-az-za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-z^{3}a^{-7}+4z^{6}a^{-6}-7z^{4}a^{-6}+2z^{2}a^{-6}+6z^{7}a^{-5}-11z^{5}a^{-5}+5z^{3}a^{-5}-za^{-5}+4z^{8}a^{-4}-11z^{4}a^{-4}+7z^{2}a^{-4}+z^{9}a^{-3}+11z^{7}a^{-3}+a^{3}z^{5}-26z^{5}a^{-3}-2a^{3}z^{3}+20z^{3}a^{-3}+a^{3}z-6za^{-3}+7z^{8}a^{-2}+3a^{2}z^{6}-6z^{6}a^{-2}-6a^{2}z^{4}-6z^{4}a^{-2}+3a^{2}z^{2}+7z^{2}a^{-2}+z^{9}a^{-1}+4az^{7}+9z^{7}a^{-1}-6az^{5}-21z^{5}a^{-1}+3az^{3}+19z^{3}a^{-1}-2az-8za^{-1}+az^{-1}+a^{-1}z^{-1}+3z^{8}+z^{6}-8z^{4}+5z^{2}-1$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

3

-3

10

4

1

3

8

6

3

-3

6

7

4

3

4

6

0

2

7

0

4

8

4

-2

2

5

-3

-4

1

4

3

-6

2

-2

-8

1

Integral Khovanov Homology

(db, data source)

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