L11a458

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Planar diagram presentation	X₆₁₇₂ X_14,4,15,3 X_16,5,17,6 X_20,9,21,10 X_18,8,19,7 X_22,19,13,20 X_8,14,9,13 X_12,15,5,16 X_10,21,11,22 X_2,11,3,12 X_4,18,1,17
Gauss code	{1, -10, 2, -11}, {3, -1, 5, -7, 4, -9, 10, -8}, {7, -2, 8, -3, 11, -5, 6, -4, 9, -6}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {uv^{3}w^{3}-2uv^{3}w^{2}+uv^{3}w-2uv^{2}w^{3}+7uv^{2}w^{2}-6uv^{2}w+uv^{2}+uvw^{3}-5uvw^{2}+7uvw-2uv+uw^{2}-2uw+u-v^{3}w^{3}+2v^{3}w^{2}-v^{3}w+2v^{2}w^{3}-7v^{2}w^{2}+5v^{2}w-v^{2}-vw^{3}+6vw^{2}-7vw+2v-w^{2}+2w-1}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$q^{3}-4q^{2}+9q-14+22q^{-1}-24q^{-2}+26q^{-3}-22q^{-4}+17q^{-5}-11q^{-6}+5q^{-7}-q^{-8}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{6}z^{4}-a^{6}z^{2}+2a^{4}z^{6}+6a^{4}z^{4}+5a^{4}z^{2}+a^{4}z^{-2}+a^{4}-a^{2}z^{8}-5a^{2}z^{6}-10a^{2}z^{4}-8a^{2}z^{2}-2a^{2}z^{-2}-3a^{2}+z^{6}+3z^{4}+3z^{2}+z^{-2}+2$ (db)
Kauffman polynomial	$2a^{4}z^{10}+2a^{2}z^{10}+8a^{5}z^{9}+14a^{3}z^{9}+6az^{9}+13a^{6}z^{8}+22a^{4}z^{8}+16a^{2}z^{8}+7z^{8}+11a^{7}z^{7}+3a^{5}z^{7}-17a^{3}z^{7}-5az^{7}+4z^{7}a^{-1}+5a^{8}z^{6}-18a^{6}z^{6}-53a^{4}z^{6}-47a^{2}z^{6}+z^{6}a^{-2}-16z^{6}+a^{9}z^{5}-15a^{7}z^{5}-24a^{5}z^{5}-7a^{3}z^{5}-8az^{5}-9z^{5}a^{-1}-4a^{8}z^{4}+6a^{6}z^{4}+40a^{4}z^{4}+45a^{2}z^{4}-2z^{4}a^{-2}+13z^{4}+4a^{7}z^{3}+13a^{5}z^{3}+12a^{3}z^{3}+8az^{3}+5z^{3}a^{-1}-a^{6}z^{2}-14a^{4}z^{2}-20a^{2}z^{2}+z^{2}a^{-2}-6z^{2}-3a^{3}z-3az+3a^{4}+5a^{2}+3+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

7

1

5

3

-3

3

6

1

5

1

8

3

-5

-1

14

6

8

-3

13

11

-2

-5

13

11

2

-7

9

13

4

-9

8

13

-5

-11

4

10

6

-13

1

7

-6

-15

4

-17

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-3$	$i=-1$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-4$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{8}$
$r=-3$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=-2$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=-1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=0$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{14}$
$r=1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$