L11n49

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Planar diagram presentation	X₆₁₇₂ X_18,7,19,8 X_4,19,1,20 X_5,14,6,15 X₈₄₉₃ X_9,16,10,17 X_15,10,16,11 X_11,20,12,21 X_13,22,14,5 X_21,12,22,13 X_2,18,3,17
Gauss code	{1, -11, 5, -3}, {-4, -1, 2, -5, -6, 7, -8, 10, -9, 4, -7, 6, 11, -2, 3, 8, -10, 9}

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Polynomial invariants

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

3

1

2

-6

3

0

-8

4

1

3

-10

2

3

1

-12

4

0

-14

1

2

1

-16

2

4

-2

-18

1

-20

2

-2

Integral Khovanov Homology

(db, data source)

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