L11a238

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Planar diagram presentation	X₈₁₉₂ X_12,4,13,3 X_22,12,7,11 X_16,9,17,10 X_14,22,15,21 X_10,15,11,16 X_18,6,19,5 X_20,18,21,17 X₂₇₃₈ X_4,14,5,13 X_6,20,1,19
Gauss code	{1, -9, 2, -10, 7, -11}, {9, -1, 4, -6, 3, -2, 10, -5, 6, -4, 8, -7, 11, -8, 5, -3}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)^{2}t(2)^{4}-2t(1)t(2)^{4}+t(2)^{4}-5t(1)^{2}t(2)^{3}+8t(1)t(2)^{3}-4t(2)^{3}+7t(1)^{2}t(2)^{2}-11t(1)t(2)^{2}+7t(2)^{2}-4t(1)^{2}t(2)+8t(1)t(2)-5t(2)+t(1)^{2}-2t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{17/2}-4q^{15/2}+9q^{13/2}-15q^{11/2}+19q^{9/2}-22q^{7/2}+21q^{5/2}-18q^{3/2}+13{\sqrt {q}}-{\frac {8}{\sqrt {q}}}+{\frac {3}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-7}+za^{-7}-2z^{5}a^{-5}-4z^{3}a^{-5}-2za^{-5}-a^{-5}z^{-1}+z^{7}a^{-3}+3z^{5}a^{-3}+4z^{3}a^{-3}+4za^{-3}+2a^{-3}z^{-1}-2z^{5}a^{-1}+az^{3}-5z^{3}a^{-1}+2az-4za^{-1}+az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-10}+4z^{5}a^{-9}-z^{3}a^{-9}+9z^{6}a^{-8}-7z^{4}a^{-8}+2z^{2}a^{-8}+14z^{7}a^{-7}-20z^{5}a^{-7}+12z^{3}a^{-7}-3za^{-7}+14z^{8}a^{-6}-21z^{6}a^{-6}+8z^{4}a^{-6}-z^{2}a^{-6}+8z^{9}a^{-5}+z^{7}a^{-5}-33z^{5}a^{-5}+29z^{3}a^{-5}-9za^{-5}+a^{-5}z^{-1}+2z^{10}a^{-4}+18z^{8}a^{-4}-55z^{6}a^{-4}+39z^{4}a^{-4}-8z^{2}a^{-4}+12z^{9}a^{-3}-22z^{7}a^{-3}-10z^{5}a^{-3}+28z^{3}a^{-3}-13za^{-3}+2a^{-3}z^{-1}+2z^{10}a^{-2}+7z^{8}a^{-2}-35z^{6}a^{-2}+34z^{4}a^{-2}-9z^{2}a^{-2}+a^{-2}+4z^{9}a^{-1}+az^{7}-8z^{7}a^{-1}-4az^{5}-5z^{5}a^{-1}+6az^{3}+18z^{3}a^{-1}-4az-11za^{-1}+az^{-1}+2a^{-1}z^{-1}+3z^{8}-10z^{6}+11z^{4}-4z^{2}$ (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

18

1

-1

16

3

14

6

1

-5

12

9

3

6

10

6

-4

8

12

9

3

6

10

11

1

4

8

11

-3

2

6

11

5

0

2

7

-5

-2

1

6

5

-4

2

-2

-6

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=2$	$i=4$
$r=-4$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=0$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{8}$
$r=1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{12}$
$r=3$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=4$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=5$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=6$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=7$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$