L11a351

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Planar diagram presentation	X_12,1,13,2 X_18,9,19,10 X_14,6,15,5 X_6,12,7,11 X_22,15,11,16 X_20,8,21,7 X₈₃₉₄ X_16,21,17,22 X_4,18,5,17 X_10,13,1,14 X_2,19,3,20
Gauss code	{1, -11, 7, -9, 3, -4, 6, -7, 2, -10}, {4, -1, 10, -3, 5, -8, 9, -2, 11, -6, 8, -5}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}t(1)^{2}-3t(2)t(1)^{2}+2t(1)^{2}-2t(2)^{2}t(1)+7t(2)t(1)-2t(1)+2t(2)^{2}-3t(2)+1\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-q^{7/2}+5q^{5/2}-13q^{3/2}+20{\sqrt {q}}-{\frac {27}{\sqrt {q}}}+{\frac {30}{q^{3/2}}}-{\frac {30}{q^{5/2}}}+{\frac {25}{q^{7/2}}}-{\frac {18}{q^{9/2}}}+{\frac {10}{q^{11/2}}}-{\frac {4}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}(-z)+3a^{5}z^{3}+2a^{5}z-3a^{3}z^{5}-5a^{3}z^{3}-3a^{3}z+az^{7}+3az^{5}-z^{5}a^{-1}+6az^{3}-z^{3}a^{-1}+4az+az^{-1}-2za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-2a^{8}z^{4}+a^{8}z^{2}+4a^{7}z^{7}-8a^{7}z^{5}+5a^{7}z^{3}-a^{7}z+8a^{6}z^{8}-16a^{6}z^{6}+12a^{6}z^{4}-4a^{6}z^{2}+9a^{5}z^{9}-13a^{5}z^{7}+3a^{5}z^{5}+2a^{5}z^{3}+4a^{4}z^{10}+14a^{4}z^{8}-44a^{4}z^{6}+33a^{4}z^{4}-7a^{4}z^{2}+22a^{3}z^{9}-36a^{3}z^{7}+9a^{3}z^{5}+z^{5}a^{-3}+3a^{3}z^{3}+4a^{2}z^{10}+24a^{2}z^{8}-59a^{2}z^{6}+5z^{6}a^{-2}+33a^{2}z^{4}-2z^{4}a^{-2}-4a^{2}z^{2}+13az^{9}-6az^{7}+13z^{7}a^{-1}-19az^{5}-16z^{5}a^{-1}+13az^{3}+7z^{3}a^{-1}-4az-3za^{-1}+az^{-1}+a^{-1}z^{-1}+18z^{8}-27z^{6}+12z^{4}-2z^{2}-1$ (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

3

4

χ

8

1

6

4

-4

4

9

1

8

2

11

4

-7

0

16

9

7

-2

16

13

-3

-4

14

0

-6

11

16

5

-8

7

14

-7

-10

3

11

8

-12

1

7

-6

-14

3

-16

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-4$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-3$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=-2$	${\mathbb {Z} }^{16}\oplus {\mathbb {Z} }_{2}^{14}$	${\mathbb {Z} }^{14}$
$r=-1$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{16}$	${\mathbb {Z} }^{16}$
$r=0$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{14}$	${\mathbb {Z} }^{16}$
$r=1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$