L11a475

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

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(2)-1)(t(3)-1)^{2}(t(3)t(1)-3t(1)-3t(3)+1)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{5}+4q^{4}-9q^{3}+15q^{2}-19q+21-20q^{-1}+18q^{-2}-11q^{-3}+7q^{-4}-2q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}z^{-2}+a^{6}-3a^{4}z^{2}-2a^{4}z^{-2}-z^{2}a^{-4}-4a^{4}+3a^{2}z^{4}+2z^{4}a^{-2}+4a^{2}z^{2}+a^{2}z^{-2}+z^{2}a^{-2}+3a^{2}-z^{6}-z^{4}-z^{2}$ (db)
Kauffman polynomial	$a^{2}z^{10}+z^{10}+3a^{3}z^{9}+8az^{9}+5z^{9}a^{-1}+3a^{4}z^{8}+11a^{2}z^{8}+9z^{8}a^{-2}+17z^{8}+2a^{5}z^{7}-3az^{7}+7z^{7}a^{-1}+8z^{7}a^{-3}+a^{6}z^{6}-3a^{4}z^{6}-27a^{2}z^{6}-11z^{6}a^{-2}+4z^{6}a^{-4}-38z^{6}-4a^{5}z^{5}-7a^{3}z^{5}-18az^{5}-28z^{5}a^{-1}-12z^{5}a^{-3}+z^{5}a^{-5}-4a^{6}z^{4}-5a^{4}z^{4}+22a^{2}z^{4}+3z^{4}a^{-2}-5z^{4}a^{-4}+31z^{4}+4a^{3}z^{3}+18az^{3}+22z^{3}a^{-1}+7z^{3}a^{-3}-z^{3}a^{-5}+6a^{6}z^{2}+9a^{4}z^{2}-7a^{2}z^{2}-2z^{2}a^{-2}+2z^{2}a^{-4}-14z^{2}+4a^{5}z+2a^{3}z-6az-6za^{-1}-2za^{-3}-4a^{6}-6a^{4}-a^{2}+a^{-2}+3-2a^{5}z^{-1}-2a^{3}z^{-1}+a^{6}z^{-2}+2a^{4}z^{-2}+a^{2}z^{-2}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

11

1

-1

9

3

7

6

1

-5

5

9

3

6

3

10

6

-4

1

11

9

2

-1

11

12

1

-3

7

9

-2

-5

4

11

7

-7

3

7

-4

-9

1

6

5

-11

1

-1

-13

1

Integral Khovanov Homology

(db, data source)

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