L11n173

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Planar diagram presentation	X₈₁₉₂ X_10,4,11,3 X_12,7,13,8 X_15,7,16,22 X_14,6,15,5 X_6,14,1,13 X_21,17,22,16 X_18,10,19,9 X_20,11,21,12 X_4,18,5,17 X_2,19,3,20
Gauss code	{1, -11, 2, -10, 5, -6}, {3, -1, 8, -2, 9, -3, 6, -5, -4, 7, 10, -8, 11, -9, -7, 4}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

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

-4

-3

-2

-1

0

1

2

3

4

5

χ

12

2

-2

10

4

8

5

2

-3

6

8

4

7

6

-1

2

7

0

5

8

3

-2

3

6

-3

-4

1

5

4

-6

3

-3

-8

1

Integral Khovanov Homology

(db, data source)

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