L11a452

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(3)^{2}t(2)^{3}-t(3)^{2}t(2)^{3}-t(1)t(3)t(2)^{3}+2t(3)t(2)^{3}-t(2)^{3}+t(1)t(3)^{3}t(2)^{2}-t(3)^{3}t(2)^{2}-2t(1)t(3)^{2}t(2)^{2}+2t(3)^{2}t(2)^{2}-t(1)t(2)^{2}+2t(1)t(3)t(2)^{2}-2t(3)t(2)^{2}+2t(2)^{2}-2t(1)t(3)^{3}t(2)+t(3)^{3}t(2)+2t(1)t(3)^{2}t(2)-2t(3)^{2}t(2)+t(1)t(2)-2t(1)t(3)t(2)+2t(3)t(2)-t(2)+t(1)t(3)^{3}-2t(1)t(3)^{2}+t(3)^{2}+t(1)t(3)-t(3)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{6}+3q^{5}-6q^{4}+9q^{3}-10q^{2}+12q-11+10q^{-1}-6q^{-2}+5q^{-3}-2q^{-4}+q^{-5}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{-4}+a^{4}z^{2}-2z^{2}a^{-4}+a^{4}z^{-2}+2a^{4}-a^{-4}+z^{6}a^{-2}-2a^{2}z^{4}+3z^{4}a^{-2}-6a^{2}z^{2}+3z^{2}a^{-2}-2a^{2}z^{-2}-5a^{2}+2a^{-2}+z^{6}+3z^{4}+3z^{2}+z^{-2}+2$ (db)
Kauffman polynomial	$a^{2}z^{10}+z^{10}+2a^{3}z^{9}+6az^{9}+4z^{9}a^{-1}+a^{4}z^{8}+a^{2}z^{8}+7z^{8}a^{-2}+7z^{8}-10a^{3}z^{7}-24az^{7}-5z^{7}a^{-1}+9z^{7}a^{-3}-6a^{4}z^{6}-24a^{2}z^{6}-13z^{6}a^{-2}+9z^{6}a^{-4}-40z^{6}+14a^{3}z^{5}+24az^{5}-12z^{5}a^{-1}-16z^{5}a^{-3}+6z^{5}a^{-5}+13a^{4}z^{4}+52a^{2}z^{4}-14z^{4}a^{-4}+3z^{4}a^{-6}+56z^{4}-2a^{3}z^{3}+az^{3}+12z^{3}a^{-1}+4z^{3}a^{-3}-4z^{3}a^{-5}+z^{3}a^{-7}-13a^{4}z^{2}-39a^{2}z^{2}+z^{2}a^{-2}+7z^{2}a^{-4}-32z^{2}-5a^{3}z-8az-3za^{-1}+za^{-3}+za^{-5}+6a^{4}+13a^{2}-a^{-4}+9+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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

13

1

-1

11

2

9

4

1

-3

7

5

2

3

5

4

-1

3

7

5

2

1

7

8

1

-1

3

4

-1

-3

3

7

4

-5

2

3

-1

-7

1

4

3

-9

1

-1

-11

1

Integral Khovanov Homology

(db, data source)

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