L11a102

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}-t(2)+1\right)\left(2t(2)^{2}-t(2)+2\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {13}{q^{9/2}}}-q^{7/2}+{\frac {17}{q^{7/2}}}+3q^{5/2}-{\frac {19}{q^{5/2}}}-7q^{3/2}+{\frac {19}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {4}{q^{13/2}}}+{\frac {8}{q^{11/2}}}+11{\sqrt {q}}-{\frac {17}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{5}z^{5}-2a^{5}z^{3}+a^{5}z^{-1}+a^{3}z^{7}+3a^{3}z^{5}+a^{3}z^{3}-4a^{3}z-3a^{3}z^{-1}+az^{7}+4az^{5}-z^{5}a^{-1}+7az^{3}-3z^{3}a^{-1}+7az-3za^{-1}+4az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2a^{4}z^{10}-2a^{2}z^{10}-5a^{5}z^{9}-10a^{3}z^{9}-5az^{9}-6a^{6}z^{8}-5a^{4}z^{8}-6a^{2}z^{8}-7z^{8}-4a^{7}z^{7}+7a^{5}z^{7}+21a^{3}z^{7}+4az^{7}-6z^{7}a^{-1}-a^{8}z^{6}+13a^{6}z^{6}+13a^{4}z^{6}+17a^{2}z^{6}-3z^{6}a^{-2}+15z^{6}+10a^{7}z^{5}-a^{5}z^{5}-17a^{3}z^{5}+9az^{5}+14z^{5}a^{-1}-z^{5}a^{-3}+2a^{8}z^{4}-5a^{6}z^{4}-12a^{2}z^{4}+5z^{4}a^{-2}-14z^{4}-6a^{7}z^{3}+6a^{5}z^{3}+3a^{3}z^{3}-25az^{3}-14z^{3}a^{-1}+2z^{3}a^{-3}-a^{8}z^{2}-a^{6}z^{2}-5a^{4}z^{2}-4a^{2}z^{2}+z^{2}-a^{7}z-2a^{5}z+7a^{3}z+15az+7za^{-1}+a^{6}+3a^{4}+3a^{2}+2-a^{5}z^{-1}-3a^{3}z^{-1}-4az^{-1}-2a^{-1}z^{-1}$ (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

8

1

6

2

-2

4

5

1

4

2

6

2

-4

0

11

5

6

-2

10

8

-2

-4

9

0

-6

8

10

2

-8

5

9

-4

-10

3

8

5

-12

1

5

-4

-14

3

-16

1

-1

Integral Khovanov Homology

(db, data source)

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