L11a459

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(3)^{3}t(2)^{3}-t(3)^{3}t(2)^{3}-2t(1)t(3)^{2}t(2)^{3}+2t(3)^{2}t(2)^{3}+t(1)t(3)t(2)^{3}-t(3)t(2)^{3}-2t(1)t(3)^{3}t(2)^{2}+2t(3)^{3}t(2)^{2}+5t(1)t(3)^{2}t(2)^{2}-5t(3)^{2}t(2)^{2}+t(1)t(2)^{2}-4t(1)t(3)t(2)^{2}+5t(3)t(2)^{2}-t(2)^{2}+t(1)t(3)^{3}t(2)-t(3)^{3}t(2)-5t(1)t(3)^{2}t(2)+4t(3)^{2}t(2)-2t(1)t(2)+5t(1)t(3)t(2)-5t(3)t(2)+2t(2)+t(1)t(3)^{2}-t(3)^{2}+t(1)-2t(1)t(3)+2t(3)-1}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{-6}-q^{5}-4q^{-5}+4q^{4}+9q^{-4}-8q^{3}-13q^{-3}+13q^{2}+19q^{-2}-18q-20q^{-1}+22$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{4}z^{4}+2a^{4}z^{2}+a^{4}z^{-2}+a^{4}-2a^{2}z^{6}-z^{6}a^{-2}-7a^{2}z^{4}-3z^{4}a^{-2}-7a^{2}z^{2}-2z^{2}a^{-2}-2a^{2}z^{-2}-3a^{2}+z^{8}+5z^{6}+9z^{4}+6z^{2}+z^{-2}+2$ (db)
Kauffman polynomial	$2a^{2}z^{10}+2z^{10}+6a^{3}z^{9}+12az^{9}+6z^{9}a^{-1}+7a^{4}z^{8}+11a^{2}z^{8}+8z^{8}a^{-2}+12z^{8}+4a^{5}z^{7}-9a^{3}z^{7}-24az^{7}-4z^{7}a^{-1}+7z^{7}a^{-3}+a^{6}z^{6}-17a^{4}z^{6}-39a^{2}z^{6}-12z^{6}a^{-2}+4z^{6}a^{-4}-37z^{6}-9a^{5}z^{5}-2a^{3}z^{5}+14az^{5}-5z^{5}a^{-1}-11z^{5}a^{-3}+z^{5}a^{-5}-2a^{6}z^{4}+12a^{4}z^{4}+40a^{2}z^{4}+7z^{4}a^{-2}-6z^{4}a^{-4}+39z^{4}+4a^{5}z^{3}+4a^{3}z^{3}+az^{3}+6z^{3}a^{-1}+4z^{3}a^{-3}-z^{3}a^{-5}+a^{6}z^{2}-7a^{4}z^{2}-18a^{2}z^{2}-3z^{2}a^{-2}+z^{2}a^{-4}-14z^{2}-3a^{3}z-3az+3a^{4}+5a^{2}+3+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

11

1

-1

9

3

7

5

1

-4

5

8

3

5

3

10

5

-5

1

12

8

4

-1

11

13

2

-3

8

9

-1

-5

5

11

6

-7

4

8

-4

-9

1

6

5

-11

3

-3

-13

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-1$	$i=1$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{4}$
$r=-3$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=0$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{12}$
$r=1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=2$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$