L11a176

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

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)^{2}t(2)^{6}-t(1)t(2)^{6}-3t(1)^{2}t(2)^{5}+4t(1)t(2)^{5}-t(2)^{5}+5t(1)^{2}t(2)^{4}-8t(1)t(2)^{4}+3t(2)^{4}-5t(1)^{2}t(2)^{3}+11t(1)t(2)^{3}-5t(2)^{3}+3t(1)^{2}t(2)^{2}-8t(1)t(2)^{2}+5t(2)^{2}-t(1)^{2}t(2)+4t(1)t(2)-3t(2)-t(1)+1}{t(1)t(2)^{3}}}$ (db)
Jones polynomial	$-q^{13/2}+4q^{11/2}-9q^{9/2}+15q^{7/2}-21q^{5/2}+23q^{3/2}-24{\sqrt {q}}+{\frac {20}{\sqrt {q}}}-{\frac {15}{q^{3/2}}}+{\frac {9}{q^{5/2}}}-{\frac {4}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-3}-4z^{5}a^{-3}-5z^{3}a^{-3}-2za^{-3}-a^{-3}z^{-1}+z^{9}a^{-1}-az^{7}+6z^{7}a^{-1}-4az^{5}+13z^{5}a^{-1}-5az^{3}+12z^{3}a^{-1}-2az+5za^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-z^{3}a^{-7}+4z^{6}a^{-6}-5z^{4}a^{-6}+z^{2}a^{-6}+8z^{7}a^{-5}-12z^{5}a^{-5}+6z^{3}a^{-5}-za^{-5}+10z^{8}a^{-4}+a^{4}z^{6}-15z^{6}a^{-4}-2a^{4}z^{4}+8z^{4}a^{-4}+a^{4}z^{2}-z^{2}a^{-4}+8z^{9}a^{-3}+4a^{3}z^{7}-8z^{7}a^{-3}-9a^{3}z^{5}+z^{5}a^{-3}+5a^{3}z^{3}+2za^{-3}-a^{-3}z^{-1}+3z^{10}a^{-2}+7a^{2}z^{8}+10z^{8}a^{-2}-15a^{2}z^{6}-27z^{6}a^{-2}+8a^{2}z^{4}+19z^{4}a^{-2}-a^{2}z^{2}-4z^{2}a^{-2}+a^{-2}+7az^{9}+15z^{9}a^{-1}-12az^{7}-32z^{7}a^{-1}+6az^{5}+29z^{5}a^{-1}-6az^{3}-18z^{3}a^{-1}+3az+6za^{-1}-a^{-1}z^{-1}+3z^{10}+7z^{8}-24z^{6}+16z^{4}-4z^{2}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

3

-3

10

6

1

5

8

9

3

-6

6

12

6

4

12

10

-2

2

12

11

1

0

9

13

4

-2

6

11

-5

-4

3

9

6

-6

1

6

-5

-8

3

-10

1

-1

Integral Khovanov Homology

(db, data source)

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