L11n174

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Planar diagram presentation	X₈₁₉₂ X_20,9,21,10 X_14,5,15,6 X_11,18,12,19 X_3,10,4,11 X_7,13,8,12 X_16,13,17,14 X_17,7,18,22 X_6,15,1,16 X_4,21,5,22 X_19,2,20,3
Gauss code	{1, 11, -5, -10, 3, -9}, {-6, -1, 2, 5, -4, 6, 7, -3, 9, -7, -8, 4, -11, -2, 10, 8}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

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

-6

-5

-4

-3

-2

-1

0

1

2

χ

0

1

-2

1

-1

-4

3

1

2

-6

2

0

-8

3

2

1

-10

2

0

-12

2

3

-1

-14

1

2

1

-16

1

2

-1

-18

2

Integral Khovanov Homology

(db, data source)

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