L11a500

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Link Presentations

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

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(w-1)\left(2uvw^{3}-4uvw^{2}+3uvw-uv-2uw^{3}+4uw^{2}-4uw-4vw^{3}+4vw^{2}-2vw-w^{4}+3w^{3}-4w^{2}+2w\right)}{{\sqrt {u}}{\sqrt {v}}w^{5/2}}}$ (db)
Jones polynomial	$-q^{8}+5q^{7}-12q^{6}+18q^{5}-23q^{4}+27q^{3}-25q^{2}+22q-14+9q^{-1}-3q^{-2}+q^{-3}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+z^{6}a^{-4}-z^{4}a^{-2}+z^{4}a^{-4}-z^{4}a^{-6}-2z^{4}+a^{2}z^{2}-7z^{2}a^{-2}+3z^{2}a^{-4}-z^{2}+a^{2}-10a^{-2}+7a^{-4}-a^{-6}+3-5a^{-2}z^{-2}+4a^{-4}z^{-2}-a^{-6}z^{-2}+2z^{-2}$ (db)
Kauffman polynomial	$z^{5}a^{-9}+5z^{6}a^{-8}-3z^{4}a^{-8}+12z^{7}a^{-7}-16z^{5}a^{-7}+5z^{3}a^{-7}-2za^{-7}+a^{-7}z^{-1}+15z^{8}a^{-6}-23z^{6}a^{-6}+11z^{4}a^{-6}-4z^{2}a^{-6}-a^{-6}z^{-2}+2a^{-6}+9z^{9}a^{-5}+3z^{7}a^{-5}-30z^{5}a^{-5}+26z^{3}a^{-5}-15za^{-5}+5a^{-5}z^{-1}+2z^{10}a^{-4}+25z^{8}a^{-4}-60z^{6}a^{-4}+47z^{4}a^{-4}-25z^{2}a^{-4}-4a^{-4}z^{-2}+14a^{-4}+14z^{9}a^{-3}-11z^{7}a^{-3}-31z^{5}a^{-3}+52z^{3}a^{-3}-33za^{-3}+9a^{-3}z^{-1}+2z^{10}a^{-2}+16z^{8}a^{-2}+a^{2}z^{6}-47z^{6}a^{-2}-3a^{2}z^{4}+53z^{4}a^{-2}+3a^{2}z^{2}-40z^{2}a^{-2}-5a^{-2}z^{-2}-a^{2}+21a^{-2}+5z^{9}a^{-1}+3az^{7}+z^{7}a^{-1}-6az^{5}-24z^{5}a^{-1}+3az^{3}+34z^{3}a^{-1}-20za^{-1}+5a^{-1}z^{-1}+6z^{8}-14z^{6}+17z^{4}-16z^{2}-2z^{-2}+9$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

17

1

-1

15

4

13

8

1

-7

11

10

4

6

9

14

9

-5

7

13

9

4

5

12

14

2

3

10

13

-3

1

6

14

8

-1

3

8

-5

-3

6

-5

1

3

-2

-7

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=1$	$i=3$
$r=-4$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=0$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{10}$
$r=1$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=2$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=3$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{14}$	${\mathbb {Z} }^{14}$
$r=4$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{10}$
$r=5$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=6$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=7$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$