L11a146

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

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)^{2}t(2)^{6}-t(1)t(2)^{6}-2t(1)^{2}t(2)^{5}+3t(1)t(2)^{5}-t(2)^{5}+2t(1)^{2}t(2)^{4}-5t(1)t(2)^{4}+2t(2)^{4}-2t(1)^{2}t(2)^{3}+5t(1)t(2)^{3}-2t(2)^{3}+2t(1)^{2}t(2)^{2}-5t(1)t(2)^{2}+2t(2)^{2}-t(1)^{2}t(2)+3t(1)t(2)-2t(2)-t(1)+1}{t(1)t(2)^{3}}}$ (db)
Jones polynomial	$-q^{13/2}+3q^{11/2}-5q^{9/2}+9q^{7/2}-12q^{5/2}+13q^{3/2}-14{\sqrt {q}}+{\frac {11}{\sqrt {q}}}-{\frac {9}{q^{3/2}}}+{\frac {5}{q^{5/2}}}-{\frac {3}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-3}-5z^{5}a^{-3}-7z^{3}a^{-3}-za^{-3}+a^{-3}z^{-1}+z^{9}a^{-1}-az^{7}+7z^{7}a^{-1}-5az^{5}+17z^{5}a^{-1}-7az^{3}+15z^{3}a^{-1}-az+za^{-1}+2az^{-1}-3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-2z^{3}a^{-7}+3z^{6}a^{-6}-7z^{4}a^{-6}+3z^{2}a^{-6}+4z^{7}a^{-5}-8z^{5}a^{-5}+3z^{3}a^{-5}+4z^{8}a^{-4}+a^{4}z^{6}-7z^{6}a^{-4}-3a^{4}z^{4}+2z^{4}a^{-4}+a^{4}z^{2}+3z^{2}a^{-4}-a^{-4}+4z^{9}a^{-3}+3a^{3}z^{7}-12z^{7}a^{-3}-10a^{3}z^{5}+20z^{5}a^{-3}+7a^{3}z^{3}-13z^{3}a^{-3}-a^{3}z+2za^{-3}+a^{-3}z^{-1}+2z^{10}a^{-2}+4a^{2}z^{8}-3z^{8}a^{-2}-12a^{2}z^{6}+7a^{2}z^{4}+7z^{4}a^{-2}-3a^{-2}+4az^{9}+8z^{9}a^{-1}-13az^{7}-32z^{7}a^{-1}+14az^{5}+53z^{5}a^{-1}-6az^{3}-31z^{3}a^{-1}-2az+za^{-1}+2az^{-1}+3a^{-1}z^{-1}+2z^{10}-3z^{8}-3z^{6}+8z^{4}-z^{2}-3$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

2

-2

10

3

1

2

8

6

2

-4

6

3

4

7

6

-1

2

7

6

1

0

5

8

3

-2

4

6

-2

-4

2

6

4

-6

1

3

-2

-8

2

-10

1

-1

Integral Khovanov Homology

(db, data source)

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