L11n193

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Planar diagram presentation	X₈₁₉₂ X_9,21,10,20 X₄₇₅₈ X_16,5,17,6 X_6,15,1,16 X_22,17,7,18 X_18,13,19,14 X_14,21,15,22 X_2,11,3,12 X_12,3,13,4 X_19,11,20,10
Gauss code	{1, -9, 10, -3, 4, -5}, {3, -1, -2, 11, 9, -10, 7, -8, 5, -4, 6, -7, -11, 2, 8, -6}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(2)^{4}-t(2)^{4}+t(1)^{2}t(2)^{3}-5t(1)t(2)^{3}+3t(2)^{3}-4t(1)^{2}t(2)^{2}+7t(1)t(2)^{2}-4t(2)^{2}+3t(1)^{2}t(2)-5t(1)t(2)+t(2)-t(1)^{2}+t(1)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-{\frac {2}{q^{3/2}}}+{\frac {5}{q^{5/2}}}-{\frac {9}{q^{7/2}}}+{\frac {11}{q^{9/2}}}-{\frac {13}{q^{11/2}}}+{\frac {12}{q^{13/2}}}-{\frac {10}{q^{15/2}}}+{\frac {7}{q^{17/2}}}-{\frac {4}{q^{19/2}}}+{\frac {1}{q^{21/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-z^{3}a^{9}+z^{5}a^{7}+z^{3}a^{7}+za^{7}+a^{7}z^{-1}+z^{5}a^{5}-2za^{5}-a^{5}z^{-1}-2z^{3}a^{3}-2za^{3}$ (db)
Kauffman polynomial	$a^{12}z^{6}-2a^{12}z^{4}+a^{12}z^{2}+4a^{11}z^{7}-11a^{11}z^{5}+7a^{11}z^{3}+5a^{10}z^{8}-12a^{10}z^{6}+4a^{10}z^{4}+2a^{10}z^{2}+2a^{9}z^{9}+4a^{9}z^{7}-20a^{9}z^{5}+12a^{9}z^{3}-2a^{9}z+9a^{8}z^{8}-19a^{8}z^{6}+8a^{8}z^{4}-a^{8}z^{2}+2a^{7}z^{9}+3a^{7}z^{7}-9a^{7}z^{5}+a^{7}z^{3}+3a^{7}z-a^{7}z^{-1}+4a^{6}z^{8}-5a^{6}z^{6}+6a^{6}z^{4}-4a^{6}z^{2}+a^{6}+3a^{5}z^{7}-a^{5}z^{3}+3a^{5}z-a^{5}z^{-1}+a^{4}z^{6}+4a^{4}z^{4}-2a^{4}z^{2}+3a^{3}z^{3}-2a^{3}z$ (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

-2

2

-4

4

1

-3

-6

5

1

4

-8

6

4

-2

-10

7

5

2

-12

5

6

1

-14

5

7

-2

-16

3

6

3

-18

1

4

-3

-20

3

-22

1

-1

Integral Khovanov Homology

(db, data source)

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