L11n46

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_5,14,6,15 X₈₄₉₃ X_9,18,10,19 X_11,20,12,21 X_13,22,14,5 X_19,10,20,11 X_21,12,22,13 X_2,16,3,15
Gauss code	{1, -11, 5, -3}, {-4, -1, 2, -5, -6, 9, -7, 10, -8, 4, 11, -2, 3, 6, -9, 7, -10, 8}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{4}-t(2)^{3}+t(2)^{2}-t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {2}{q^{5/2}}}+{\frac {2}{q^{7/2}}}-{\frac {5}{q^{9/2}}}+{\frac {5}{q^{11/2}}}-{\frac {7}{q^{13/2}}}+{\frac {7}{q^{15/2}}}-{\frac {5}{q^{17/2}}}+{\frac {4}{q^{19/2}}}-{\frac {2}{q^{21/2}}}+{\frac {1}{q^{23/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{9}\left(-z^{5}\right)-4a^{9}z^{3}-5a^{9}z-3a^{9}z^{-1}+a^{7}z^{7}+6a^{7}z^{5}+14a^{7}z^{3}+16a^{7}z+7a^{7}z^{-1}-2a^{5}z^{5}-9a^{5}z^{3}-11a^{5}z-4a^{5}z^{-1}$ (db)
Kauffman polynomial	$a^{14}z^{4}-2a^{14}z^{2}+a^{14}+2a^{13}z^{5}-3a^{13}z^{3}+2a^{12}z^{6}-a^{12}z^{4}-2a^{12}z^{2}+2a^{11}z^{7}-2a^{11}z^{5}+a^{11}z^{3}+2a^{10}z^{8}-5a^{10}z^{6}+7a^{10}z^{4}-a^{10}z^{2}+a^{9}z^{9}-2a^{9}z^{7}+3a^{9}z^{5}-3a^{9}z^{3}+6a^{9}z-3a^{9}z^{-1}+3a^{8}z^{8}-11a^{8}z^{6}+18a^{8}z^{4}-16a^{8}z^{2}+7a^{8}+a^{7}z^{9}-4a^{7}z^{7}+10a^{7}z^{5}-19a^{7}z^{3}+18a^{7}z-7a^{7}z^{-1}+a^{6}z^{8}-4a^{6}z^{6}+9a^{6}z^{4}-15a^{6}z^{2}+7a^{6}+3a^{5}z^{5}-12a^{5}z^{3}+12a^{5}z-4a^{5}z^{-1}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-4

2

-6

2

0

-8

3

-10

2

0

-12

5

3

2

-14

2

0

-16

3

5

-2

-18

1

2

1

-20

1

3

-2

-22

1

-24

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-6$	$i=-4$
$r=-9$	${\mathbb {Z} }$
$r=-8$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-5$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-4$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-2$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-1$	${\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=0$	${\mathbb {Z} }^{2}$	${\mathbb {Z} }^{2}$