L11n112

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_7,18,8,19 X_19,22,20,5 X_9,21,10,20 X_21,9,22,8 X_11,17,12,16 X_17,15,18,14 X_15,11,16,10 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 $u$ , $v$ , $w$ , ...)	$-{\frac {2(u-1)(v-1)}{{\sqrt {u}}{\sqrt {v}}}}$ (db)
Jones polynomial	$-4q^{9/2}+3q^{7/2}-3q^{5/2}-{\frac {1}{q^{5/2}}}+2q^{3/2}+{\frac {1}{q^{3/2}}}+q^{15/2}-2q^{13/2}+3q^{11/2}-{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$za^{-7}+a^{-7}z^{-1}-2z^{3}a^{-5}-5za^{-5}-4a^{-5}z^{-1}+z^{5}a^{-3}+5z^{3}a^{-3}+10za^{-3}+6a^{-3}z^{-1}-z^{5}a^{-1}+az^{3}-6z^{3}a^{-1}+3az-9za^{-1}+2az^{-1}-5a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{8}a^{-2}-z^{8}a^{-4}-z^{8}a^{-6}-z^{8}-az^{7}-2z^{7}a^{-1}-3z^{7}a^{-3}-4z^{7}a^{-5}-2z^{7}a^{-7}+7z^{6}a^{-2}+3z^{6}a^{-4}+z^{6}a^{-6}-z^{6}a^{-8}+6z^{6}+6az^{5}+16z^{5}a^{-1}+20z^{5}a^{-3}+18z^{5}a^{-5}+8z^{5}a^{-7}-10z^{4}a^{-2}+2z^{4}a^{-4}+8z^{4}a^{-6}+4z^{4}a^{-8}-8z^{4}-10az^{3}-33z^{3}a^{-1}-39z^{3}a^{-3}-24z^{3}a^{-5}-8z^{3}a^{-7}-7z^{2}a^{-4}-9z^{2}a^{-6}-4z^{2}a^{-8}+2z^{2}+7az+22za^{-1}+27za^{-3}+16za^{-5}+4za^{-7}+a^{-2}+3a^{-4}+3a^{-6}+a^{-8}+1-2az^{-1}-5a^{-1}z^{-1}-6a^{-3}z^{-1}-4a^{-5}z^{-1}-a^{-7}z^{-1}$ (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

1

12

2

1

-1

10

2

1

8

1

2

1

6

2

0

4

1

2

1

2

1

2

-1

0

1

3

2

-2

1

0

-4

0

-6

1

Integral Khovanov Homology

(db, data source)

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