L11a266

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_12,4,13,3 X_22,12,9,11 X_2,9,3,10 X_4,22,5,21 X_14,6,15,5 X_16,20,17,19 X_18,8,19,7 X_6,18,7,17 X_20,16,21,15 X_8,14,1,13
Gauss code	{1, -4, 2, -5, 6, -9, 8, -11}, {4, -1, 3, -2, 11, -6, 10, -7, 9, -8, 7, -10, 5, -3}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2t(2)^{2}t(1)^{3}-3t(2)t(1)^{3}+t(1)^{3}+2t(2)^{3}t(1)^{2}-9t(2)^{2}t(1)^{2}+9t(2)t(1)^{2}-3t(1)^{2}-3t(2)^{3}t(1)+9t(2)^{2}t(1)-9t(2)t(1)+2t(1)+t(2)^{3}-3t(2)^{2}+2t(2)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$18q^{9/2}-16q^{7/2}+11q^{5/2}-7q^{3/2}+q^{21/2}-4q^{19/2}+8q^{17/2}-12q^{15/2}+16q^{13/2}-19q^{11/2}+3{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-9}-z^{5}a^{-7}+za^{-7}-2z^{5}a^{-5}-3z^{3}a^{-5}-2za^{-5}-a^{-5}z^{-1}-z^{5}a^{-3}+2za^{-3}+a^{-3}z^{-1}+z^{3}a^{-1}+za^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-12}-2z^{4}a^{-12}+z^{2}a^{-12}+4z^{7}a^{-11}-10z^{5}a^{-11}+6z^{3}a^{-11}-za^{-11}+6z^{8}a^{-10}-14z^{6}a^{-10}+7z^{4}a^{-10}+4z^{9}a^{-9}-z^{7}a^{-9}-15z^{5}a^{-9}+13z^{3}a^{-9}-3za^{-9}+z^{10}a^{-8}+11z^{8}a^{-8}-28z^{6}a^{-8}+17z^{4}a^{-8}-2z^{2}a^{-8}+7z^{9}a^{-7}-4z^{7}a^{-7}-16z^{5}a^{-7}+19z^{3}a^{-7}-5za^{-7}+z^{10}a^{-6}+10z^{8}a^{-6}-20z^{6}a^{-6}+12z^{4}a^{-6}-z^{2}a^{-6}+3z^{9}a^{-5}+6z^{7}a^{-5}-20z^{5}a^{-5}+20z^{3}a^{-5}-8za^{-5}+a^{-5}z^{-1}+5z^{8}a^{-4}-4z^{6}a^{-4}-z^{4}a^{-4}+2z^{2}a^{-4}-a^{-4}+5z^{7}a^{-3}-8z^{5}a^{-3}+6z^{3}a^{-3}-4za^{-3}+a^{-3}z^{-1}+3z^{6}a^{-2}-5z^{4}a^{-2}+2z^{2}a^{-2}+z^{5}a^{-1}-2z^{3}a^{-1}+za^{-1}$ (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		\

-2

-1

0

1

2

3

4

5

6

7

8

9

χ

22

1

-1

20

3

18

5

1

-4

16

7

3

4

14

9

5

-4

12

10

7

3

10

9

10

1

8

7

9

-2

6

4

9

5

4

3

7

-4

2

1

5

4

0

2

-2

1

Integral Khovanov Homology

(db, data source)

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