L11a349

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Planar diagram presentation	X_12,1,13,2 X₈₄₉₃ X_14,6,15,5 X_18,8,19,7 X_20,9,21,10 X_10,11,1,12 X_6,14,7,13 X_4,18,5,17 X_22,15,11,16 X_2,19,3,20 X_16,21,17,22
Gauss code	{1, -10, 2, -8, 3, -7, 4, -2, 5, -6}, {6, -1, 7, -3, 9, -11, 8, -4, 10, -5, 11, -9}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}t(1)^{2}-3t(2)t(1)^{2}+t(1)^{2}-3t(2)^{2}t(1)+5t(2)t(1)-3t(1)+t(2)^{2}-3t(2)+1\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-4q^{7/2}+10q^{5/2}-18q^{3/2}+23{\sqrt {q}}-{\frac {28}{\sqrt {q}}}+{\frac {27}{q^{3/2}}}-{\frac {24}{q^{5/2}}}+{\frac {18}{q^{7/2}}}-{\frac {10}{q^{9/2}}}+{\frac {4}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+a^{5}z-2a^{3}z^{5}-3a^{3}z^{3}+z^{3}a^{-3}-a^{3}z+za^{-3}+az^{7}+2az^{5}-2z^{5}a^{-1}+az^{3}-3z^{3}a^{-1}-za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-3a^{2}z^{10}-3z^{10}-10a^{3}z^{9}-18az^{9}-8z^{9}a^{-1}-13a^{4}z^{8}-22a^{2}z^{8}-8z^{8}a^{-2}-17z^{8}-9a^{5}z^{7}+5a^{3}z^{7}+26az^{7}+8z^{7}a^{-1}-4z^{7}a^{-3}-4a^{6}z^{6}+21a^{4}z^{6}+58a^{2}z^{6}+16z^{6}a^{-2}-z^{6}a^{-4}+50z^{6}-a^{7}z^{5}+12a^{5}z^{5}+13a^{3}z^{5}+8z^{5}a^{-1}+8z^{5}a^{-3}+4a^{6}z^{4}-17a^{4}z^{4}-46a^{2}z^{4}-11z^{4}a^{-2}+2z^{4}a^{-4}-38z^{4}+a^{7}z^{3}-8a^{5}z^{3}-12a^{3}z^{3}-5az^{3}-8z^{3}a^{-1}-6z^{3}a^{-3}-a^{6}z^{2}+6a^{4}z^{2}+14a^{2}z^{2}+3z^{2}a^{-2}-z^{2}a^{-4}+11z^{2}+2a^{5}z+2a^{3}z+2za^{-1}+2za^{-3}+1-az^{-1}-a^{-1}z^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

3

6

7

1

-6

4

11

3

8

2

12

7

-5

0

16

11

5

-2

13

14

1

-4

11

14

-3

-6

7

13

6

-8

3

11

-8

-10

1

7

6

-12

3

-3

-14

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-3$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-2$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=-1$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=0$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{14}$	${\mathbb {Z} }^{16}$
$r=1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=2$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$