L11a400

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,6,13,5 X₈₄₉₃ X_2,14,3,13 X_14,7,15,8 X_18,12,19,11 X_20,9,21,10 X_10,19,5,20 X_4,15,1,16 X_22,18,11,17 X_16,22,17,21
Gauss code	{1, -4, 3, -9}, {2, -1, 5, -3, 7, -8}, {6, -2, 4, -5, 9, -11, 10, -6, 8, -7, 11, -10}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)^{2}(w-1)^{2}\left(w^{2}+1\right)}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$q^{7}-4q^{6}+8q^{5}-14q^{4}+18q^{3}-20q^{2}+21q-16+14q^{-1}-7q^{-2}+4q^{-3}-q^{-4}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{8}a^{-2}-5z^{6}a^{-2}+z^{6}a^{-4}+2z^{6}-a^{2}z^{4}-9z^{4}a^{-2}+3z^{4}a^{-4}+7z^{4}-2a^{2}z^{2}-5z^{2}a^{-2}+2z^{2}a^{-4}+5z^{2}+a^{2}+4a^{-2}-a^{-4}-4+2a^{2}z^{-2}+4a^{-2}z^{-2}-a^{-4}z^{-2}-5z^{-2}$ (db)
Kauffman polynomial	$2z^{10}a^{-2}+2z^{10}+5az^{9}+13z^{9}a^{-1}+8z^{9}a^{-3}+4a^{2}z^{8}+18z^{8}a^{-2}+13z^{8}a^{-4}+9z^{8}+a^{3}z^{7}-14az^{7}-32z^{7}a^{-1}-5z^{7}a^{-3}+12z^{7}a^{-5}-15a^{2}z^{6}-67z^{6}a^{-2}-24z^{6}a^{-4}+8z^{6}a^{-6}-50z^{6}-3a^{3}z^{5}+7az^{5}+11z^{5}a^{-1}-19z^{5}a^{-3}-16z^{5}a^{-5}+4z^{5}a^{-7}+18a^{2}z^{4}+68z^{4}a^{-2}+18z^{4}a^{-4}-6z^{4}a^{-6}+z^{4}a^{-8}+61z^{4}+2a^{3}z^{3}+az^{3}+6z^{3}a^{-1}+15z^{3}a^{-3}+6z^{3}a^{-5}-2z^{3}a^{-7}-7a^{2}z^{2}-24z^{2}a^{-2}-8z^{2}a^{-4}-23z^{2}+5az+9za^{-1}+5za^{-3}+za^{-5}-3a^{2}-2a^{-2}-4-5az^{-1}-9a^{-1}z^{-1}-5a^{-3}z^{-1}-a^{-5}z^{-1}+2a^{2}z^{-2}+4a^{-2}z^{-2}+a^{-4}z^{-2}+5z^{-2}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

15

1

13

3

-3

11

5

1

4

9

3

-6

7

9

5

4

5

11

9

-2

3

10

9

1

8

13

5

-1

6

8

-2

-3

3

10

7

-5

1

4

-3

-7

3

-9

1

-1

Integral Khovanov Homology

(db, data source)

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