L11n94

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_9,14,10,15 X_19,22,20,5 X_11,21,12,20 X_21,11,22,10 X_15,19,16,18 X_7,17,8,16 X_17,9,18,8 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, -8, 9, -3, 6, -5, -2, 11, 3, -7, 8, -9, 7, -4, 5, -6, 4}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)\left(v^{2}+1\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-2q^{7/2}+2q^{5/2}-{\frac {1}{q^{5/2}}}-4q^{3/2}+{\frac {2}{q^{3/2}}}+q^{15/2}-q^{13/2}+3{\sqrt {q}}-{\frac {3}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$za^{-7}+a^{-7}z^{-1}-z^{3}a^{-5}-4za^{-5}-3a^{-5}z^{-1}+2z^{3}a^{-3}+6za^{-3}+3a^{-3}z^{-1}-z^{5}a^{-1}+az^{3}-4z^{3}a^{-1}+2az-5za^{-1}+az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{9}a^{-1}-z^{9}a^{-3}-4z^{8}a^{-2}-2z^{8}a^{-4}-2z^{8}-az^{7}+2z^{7}a^{-1}+3z^{7}a^{-3}-z^{7}a^{-5}-z^{7}a^{-7}+21z^{6}a^{-2}+12z^{6}a^{-4}-z^{6}a^{-8}+10z^{6}+5az^{5}+9z^{5}a^{-1}+6z^{5}a^{-3}+8z^{5}a^{-5}+6z^{5}a^{-7}-29z^{4}a^{-2}-18z^{4}a^{-4}+3z^{4}a^{-6}+5z^{4}a^{-8}-13z^{4}-7az^{3}-19z^{3}a^{-1}-18z^{3}a^{-3}-14z^{3}a^{-5}-8z^{3}a^{-7}+13z^{2}a^{-2}+8z^{2}a^{-4}-5z^{2}a^{-6}-5z^{2}a^{-8}+5z^{2}+4az+11za^{-1}+14za^{-3}+10za^{-5}+3za^{-7}-2a^{-2}+2a^{-6}+a^{-8}-az^{-1}-2a^{-1}z^{-1}-3a^{-3}z^{-1}-3a^{-5}z^{-1}-a^{-7}z^{-1}$ (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

0

12

1

10

2

1

-1

8

2

1

6

2

3

1

0

4

3

2

1

2

4

1

0

1

2

1

0

-2

1

2

1

-4

1

-1

-6

1

Integral Khovanov Homology

(db, data source)

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