L11n81

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Planar diagram presentation	X₆₁₇₂ X_18,7,19,8 X_20,9,21,10 X_8,19,9,20 X_4,21,1,22 X_5,14,6,15 X_10,4,11,3 X_11,16,12,17 X_15,12,16,13 X_13,22,14,5 X_2,18,3,17
Gauss code	{1, -11, 7, -5}, {-6, -1, 2, -4, 3, -7, -8, 9, -10, 6, -9, 8, 11, -2, 4, -3, 5, 10}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)\left(v^{2}+1\right)\left(v^{2}-v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {8}{q^{9/2}}}+{\frac {5}{q^{7/2}}}-{\frac {5}{q^{5/2}}}+{\frac {2}{q^{3/2}}}+{\frac {2}{q^{19/2}}}-{\frac {4}{q^{17/2}}}+{\frac {6}{q^{15/2}}}-{\frac {7}{q^{13/2}}}+{\frac {8}{q^{11/2}}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-za^{9}-a^{9}z^{-1}-z^{5}a^{7}-2z^{3}a^{7}+za^{7}+a^{7}z^{-1}+z^{7}a^{5}+5z^{5}a^{5}+8z^{3}a^{5}+5za^{5}+2a^{5}z^{-1}-z^{5}a^{3}-4z^{3}a^{3}-5za^{3}-2a^{3}z^{-1}$ (db)
Kauffman polynomial	$-3z^{2}a^{12}+a^{12}-z^{5}a^{11}-3z^{3}a^{11}+za^{11}-3z^{6}a^{10}+3z^{4}a^{10}-2z^{2}a^{10}-4z^{7}a^{9}+8z^{5}a^{9}-6z^{3}a^{9}+2za^{9}+a^{9}z^{-1}-3z^{8}a^{8}+5z^{6}a^{8}-z^{4}a^{8}+6z^{2}a^{8}-3a^{8}-z^{9}a^{7}-4z^{7}a^{7}+21z^{5}a^{7}-19z^{3}a^{7}+4za^{7}+a^{7}z^{-1}-5z^{8}a^{6}+16z^{6}a^{6}-11z^{4}a^{6}+2z^{2}a^{6}-z^{9}a^{5}-z^{7}a^{5}+17z^{5}a^{5}-25z^{3}a^{5}+10za^{5}-2a^{5}z^{-1}-2z^{8}a^{4}+8z^{6}a^{4}-7z^{4}a^{4}-3z^{2}a^{4}+3a^{4}-z^{7}a^{3}+5z^{5}a^{3}-9z^{3}a^{3}+7za^{3}-2a^{3}z^{-1}$ (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

χ

0

1

-2

1

-1

-4

4

1

3

-6

3

0

-8

5

2

3

-10

3

0

-12

4

5

-1

-14

2

3

1

-16

2

4

-2

-18

2

-20

2

-2

Integral Khovanov Homology

(db, data source)

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