L11a178

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Planar diagram presentation	X₈₁₉₂ X_10,4,11,3 X_16,7,17,8 X_20,10,21,9 X_22,15,7,16 X_14,6,15,5 X_18,14,19,13 X_12,20,13,19 X_4,12,5,11 X_6,18,1,17 X_2,21,3,22
Gauss code	{1, -11, 2, -9, 6, -10}, {3, -1, 4, -2, 9, -8, 7, -6, 5, -3, 10, -7, 8, -4, 11, -5}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2u^{2}v^{4}-5u^{2}v^{3}+6u^{2}v^{2}-4u^{2}v+u^{2}-3uv^{4}+10uv^{3}-15uv^{2}+10uv-3u+v^{4}-4v^{3}+6v^{2}-5v+2}{uv^{2}}}$ (db)
Jones polynomial	$-16q^{9/2}+21q^{7/2}-{\frac {1}{q^{7/2}}}-25q^{5/2}+{\frac {4}{q^{5/2}}}+25q^{3/2}-{\frac {10}{q^{3/2}}}+q^{15/2}-4q^{13/2}+9q^{11/2}-22{\sqrt {q}}+{\frac {16}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}-z^{7}a^{-3}+az^{5}-3z^{5}a^{-1}-3z^{5}a^{-3}+z^{5}a^{-5}+2az^{3}-4z^{3}a^{-1}-3z^{3}a^{-3}+2z^{3}a^{-5}+2az-3za^{-1}+za^{-3}+za^{-5}+az^{-1}-2a^{-1}z^{-1}+2a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$-3z^{10}a^{-2}-3z^{10}a^{-4}-9z^{9}a^{-1}-16z^{9}a^{-3}-7z^{9}a^{-5}-15z^{8}a^{-2}-10z^{8}a^{-4}-7z^{8}a^{-6}-12z^{8}-9az^{7}+7z^{7}a^{-1}+29z^{7}a^{-3}+9z^{7}a^{-5}-4z^{7}a^{-7}-4a^{2}z^{6}+42z^{6}a^{-2}+32z^{6}a^{-4}+14z^{6}a^{-6}-z^{6}a^{-8}+21z^{6}-a^{3}z^{5}+14az^{5}+9z^{5}a^{-1}-12z^{5}a^{-3}+3z^{5}a^{-5}+9z^{5}a^{-7}+4a^{2}z^{4}-34z^{4}a^{-2}-23z^{4}a^{-4}-7z^{4}a^{-6}+2z^{4}a^{-8}-16z^{4}+a^{3}z^{3}-9az^{3}-16z^{3}a^{-1}-5z^{3}a^{-3}-5z^{3}a^{-5}-6z^{3}a^{-7}+9z^{2}a^{-2}+6z^{2}a^{-4}+z^{2}a^{-6}-z^{2}a^{-8}+5z^{2}+4az+9za^{-1}+7za^{-3}+3za^{-5}+za^{-7}-a^{-2}-az^{-1}-2a^{-1}z^{-1}-2a^{-3}z^{-1}-a^{-5}z^{-1}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

16

1

-1

14

3

12

6

1

-5

10

3

7

8

11

6

-5

6

14

10

4

12

0

2

10

13

-3

0

7

13

6

-2

3

9

-6

-4

1

7

6

-6

3

-3

-8

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-4$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=0$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{10}$
$r=1$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=2$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{14}$
$r=3$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=4$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=5$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=6$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=7$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$