L11a408

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

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)t(3)^{2}t(2)^{2}-2t(3)^{2}t(2)^{2}+2t(1)t(2)^{2}-3t(1)t(3)t(2)^{2}+5t(3)t(2)^{2}-2t(2)^{2}-3t(1)t(3)^{2}t(2)+5t(3)^{2}t(2)-5t(1)t(2)+8t(1)t(3)t(2)-8t(3)t(2)+3t(2)+2t(1)t(3)^{2}-2t(3)^{2}+2t(1)-5t(1)t(3)+3t(3)-1}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$-q^{5}+4q^{4}-8q^{3}+14q^{2}-17q+21-19q^{-1}+17q^{-2}-12q^{-3}+7q^{-4}-3q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}-3z^{2}a^{4}-2a^{4}+3z^{4}a^{2}+4z^{2}a^{2}+a^{2}z^{-2}+3a^{2}-z^{6}-2z^{4}-5z^{2}-2z^{-2}-4+2z^{4}a^{-2}+2z^{2}a^{-2}+a^{-2}z^{-2}+2a^{-2}-z^{2}a^{-4}$ (db)
Kauffman polynomial	$a^{6}z^{6}-3a^{6}z^{4}+3a^{6}z^{2}-a^{6}+3a^{5}z^{7}-8a^{5}z^{5}+z^{5}a^{-5}+7a^{5}z^{3}-z^{3}a^{-5}-2a^{5}z+4a^{4}z^{8}-6a^{4}z^{6}+4z^{6}a^{-4}-2a^{4}z^{4}-6z^{4}a^{-4}+5a^{4}z^{2}+3z^{2}a^{-4}-a^{4}+3a^{3}z^{9}+3a^{3}z^{7}+7z^{7}a^{-3}-19a^{3}z^{5}-10z^{5}a^{-3}+18a^{3}z^{3}+4z^{3}a^{-3}-6a^{3}z+a^{2}z^{10}+10a^{2}z^{8}+7z^{8}a^{-2}-20a^{2}z^{6}-5z^{6}a^{-2}+5a^{2}z^{4}-5z^{4}a^{-2}+6a^{2}z^{2}+6z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-3a^{2}-4a^{-2}+7az^{9}+4z^{9}a^{-1}+7z^{7}a^{-1}-20az^{5}-20z^{5}a^{-1}+14az^{3}+8z^{3}a^{-1}+4za^{-1}-2az^{-1}-2a^{-1}z^{-1}+z^{10}+13z^{8}-22z^{6}+5z^{4}+7z^{2}+2z^{-2}-6$ (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

11

1

-1

9

3

7

5

1

-4

5

9

3

6

3

10

7

-3

1

11

7

4

-1

9

11

2

-3

8

10

-2

-5

4

9

5

-7

3

8

-5

-9

1

5

4

-11

2

-2

-13

1

Integral Khovanov Homology

(db, data source)

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