L11a91

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_18,8,19,7 X_22,20,5,19 X_20,9,21,10 X_8,21,9,22 X_16,11,17,12 X_14,17,15,18 X_10,15,11,16 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -6, 5, -9, 7, -2, 11, -8, 9, -7, 8, -3, 4, -5, 6, -4}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)\left(v^{4}-4v^{3}+7v^{2}-4v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-q^{7/2}+4q^{5/2}-10q^{3/2}+15{\sqrt {q}}-{\frac {20}{\sqrt {q}}}+{\frac {22}{q^{3/2}}}-{\frac {22}{q^{5/2}}}+{\frac {18}{q^{7/2}}}-{\frac {13}{q^{9/2}}}+{\frac {7}{q^{11/2}}}-{\frac {3}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}-a^{7}z^{-1}+3z^{3}a^{5}+6za^{5}+4a^{5}z^{-1}-3z^{5}a^{3}-9z^{3}a^{3}-12za^{3}-6a^{3}z^{-1}+z^{7}a+4z^{5}a+9z^{3}a+10za+5az^{-1}-z^{5}a^{-1}-2z^{3}a^{-1}-3za^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{4}z^{10}-a^{2}z^{10}-3a^{5}z^{9}-8a^{3}z^{9}-5az^{9}-4a^{6}z^{8}-13a^{4}z^{8}-19a^{2}z^{8}-10z^{8}-3a^{7}z^{7}-6a^{5}z^{7}-6a^{3}z^{7}-12az^{7}-9z^{7}a^{-1}-a^{8}z^{6}+5a^{6}z^{6}+27a^{4}z^{6}+38a^{2}z^{6}-4z^{6}a^{-2}+13z^{6}+8a^{7}z^{5}+29a^{5}z^{5}+50a^{3}z^{5}+45az^{5}+15z^{5}a^{-1}-z^{5}a^{-3}+3a^{8}z^{4}+4a^{6}z^{4}-10a^{4}z^{4}-19a^{2}z^{4}+4z^{4}a^{-2}-4z^{4}-8a^{7}z^{3}-34a^{5}z^{3}-60a^{3}z^{3}-47az^{3}-12z^{3}a^{-1}+z^{3}a^{-3}-3a^{8}z^{2}-8a^{6}z^{2}-6a^{4}z^{2}-z^{2}a^{-2}+4a^{7}z+18a^{5}z+31a^{3}z+24az+7za^{-1}+a^{8}+3a^{6}+3a^{4}+a^{2}+1-a^{7}z^{-1}-4a^{5}z^{-1}-6a^{3}z^{-1}-5az^{-1}-2a^{-1}z^{-1}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

3

-3

4

7

1

6

2

8

3

-5

0

12

7

5

-2

12

10

-2

-4

10

0

-6

8

12

4

-8

5

10

-5

-10

2

8

6

-12

1

5

-4

-14

2

-16

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-4$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-3$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-2$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=0$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{12}$
$r=1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$