L11a488

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Planar diagram presentation	X₆₁₇₂ X_10,4,11,3 X_12,8,13,7 X_16,10,17,9 X_22,18,19,17 X_20,14,21,13 X_14,20,15,19 X_18,22,5,21 X_8,16,9,15 X₂₅₃₆ X_4,12,1,11
Gauss code	{1, -10, 2, -11}, {7, -6, 8, -5}, {10, -1, 3, -9, 4, -2, 11, -3, 6, -7, 9, -4, 5, -8}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(w-1)^{2}\left(2vw^{2}-vw+v+w^{3}-w^{2}+2w\right)}{{\sqrt {u}}{\sqrt {v}}w^{5/2}}}$ (db)
Jones polynomial	$-q^{11}+3q^{10}-8q^{9}+14q^{8}-18q^{7}+21q^{6}-20q^{5}+19q^{4}-12q^{3}+8q^{2}-3q+1$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$-z^{2}a^{-10}-a^{-10}z^{-2}-2a^{-10}+3z^{4}a^{-8}+8z^{2}a^{-8}+4a^{-8}z^{-2}+9a^{-8}-2z^{6}a^{-6}-7z^{4}a^{-6}-12z^{2}a^{-6}-5a^{-6}z^{-2}-13a^{-6}-z^{6}a^{-4}-z^{4}a^{-4}+3z^{2}a^{-4}+2a^{-4}z^{-2}+5a^{-4}+z^{4}a^{-2}+2z^{2}a^{-2}+a^{-2}$ (db)
Kauffman polynomial	$z^{10}a^{-6}+z^{10}a^{-8}+4z^{9}a^{-5}+9z^{9}a^{-7}+5z^{9}a^{-9}+5z^{8}a^{-4}+16z^{8}a^{-6}+19z^{8}a^{-8}+8z^{8}a^{-10}+3z^{7}a^{-3}-5z^{7}a^{-7}+4z^{7}a^{-9}+6z^{7}a^{-11}+z^{6}a^{-2}-11z^{6}a^{-4}-51z^{6}a^{-6}-56z^{6}a^{-8}-14z^{6}a^{-10}+3z^{6}a^{-12}-7z^{5}a^{-3}-18z^{5}a^{-5}-29z^{5}a^{-7}-28z^{5}a^{-9}-9z^{5}a^{-11}+z^{5}a^{-13}-3z^{4}a^{-2}+10z^{4}a^{-4}+64z^{4}a^{-6}+71z^{4}a^{-8}+16z^{4}a^{-10}-4z^{4}a^{-12}+4z^{3}a^{-3}+24z^{3}a^{-5}+50z^{3}a^{-7}+39z^{3}a^{-9}+7z^{3}a^{-11}-2z^{3}a^{-13}+3z^{2}a^{-2}-11z^{2}a^{-4}-48z^{2}a^{-6}-48z^{2}a^{-8}-13z^{2}a^{-10}+z^{2}a^{-12}-17za^{-5}-35za^{-7}-23za^{-9}-4za^{-11}+za^{-13}-a^{-2}+8a^{-4}+22a^{-6}+19a^{-8}+5a^{-10}+5a^{-5}z^{-1}+9a^{-7}z^{-1}+5a^{-9}z^{-1}+a^{-11}z^{-1}-2a^{-4}z^{-2}-5a^{-6}z^{-2}-4a^{-8}z^{-2}-a^{-10}z^{-2}$ (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		\

-2

-1

0

1

2

3

4

5

6

7

8

9

χ

23

1

-1

21

2

19

6

1

-5

17

8

2

6

15

10

6

-4

13

11

8

3

11

12

1

9

8

9

-1

7

4

11

7

5

4

8

-4

3

1

6

5

1

2

-2

-1

1

Integral Khovanov Homology

(db, data source)

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