L11n126

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Planar diagram presentation	X₈₁₉₂ X_10,4,11,3 X_22,10,7,9 X₂₇₃₈ X_15,5,16,4 X_5,13,6,12 X_11,16,12,17 X_6,18,1,17 X_14,20,15,19 X_20,14,21,13 X_18,21,19,22
Gauss code	{1, -4, 2, 5, -6, -8}, {4, -1, 3, -2, -7, 6, 10, -9, -5, 7, 8, -11, 9, -10, 11, -3}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(2)^{4}-t(2)^{4}-4t(1)t(2)^{3}+3t(2)^{3}-2t(1)^{2}t(2)^{2}+7t(1)t(2)^{2}-2t(2)^{2}+3t(1)^{2}t(2)-4t(1)t(2)-t(1)^{2}+t(1)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{15/2}-3q^{13/2}+5q^{11/2}-8q^{9/2}+9q^{7/2}-10q^{5/2}+9q^{3/2}-7{\sqrt {q}}+{\frac {4}{\sqrt {q}}}-{\frac {2}{q^{3/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-3}-3z^{3}a^{-1}+2z^{3}a^{-3}-2z^{3}a^{-5}+2az-4za^{-1}+4za^{-3}-2za^{-5}+za^{-7}+az^{-1}-2a^{-1}z^{-1}+2a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$-z^{9}a^{-3}-z^{9}a^{-5}-3z^{8}a^{-2}-6z^{8}a^{-4}-3z^{8}a^{-6}-3z^{7}a^{-1}-5z^{7}a^{-3}-5z^{7}a^{-5}-3z^{7}a^{-7}+6z^{6}a^{-2}+15z^{6}a^{-4}+7z^{6}a^{-6}-z^{6}a^{-8}-z^{6}+7z^{5}a^{-1}+20z^{5}a^{-3}+23z^{5}a^{-5}+10z^{5}a^{-7}-8z^{4}a^{-2}-10z^{4}a^{-4}-z^{4}a^{-6}+3z^{4}a^{-8}-2z^{4}-3az^{3}-15z^{3}a^{-1}-26z^{3}a^{-3}-23z^{3}a^{-5}-9z^{3}a^{-7}+3z^{2}a^{-2}+2z^{2}a^{-4}-z^{2}a^{-6}-2z^{2}a^{-8}+2z^{2}+4az+11za^{-1}+13za^{-3}+9za^{-5}+3za^{-7}-a^{-2}-az^{-1}-2a^{-1}z^{-1}-2a^{-3}z^{-1}-a^{-5}z^{-1}$ (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		\

-2

-1

0

1

2

3

4

5

6

7

χ

16

1

-1

14

2

12

3

1

-2

10

5

2

3

8

4

3

-1

6

5

1

4

5

1

2

3

5

-2

0

2

5

3

-2

2

-2

-4

2

Integral Khovanov Homology

(db, data source)

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