L11n106

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_7,16,8,17 X_17,22,18,5 X_13,18,14,19 X_9,21,10,20 X_19,14,20,15 X_21,9,22,8 X_15,10,16,11 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 8, -6, 9, 11, -2, -5, 7, -9, 3, -4, 5, -7, 6, -8, 4}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{5}-2uv^{4}-uv^{3}+3uv^{2}-2uv-2v^{4}+3v^{3}-v^{2}-2v+1}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{9/2}}}-{\frac {3}{q^{11/2}}}+{\frac {2}{q^{13/2}}}-{\frac {3}{q^{15/2}}}+{\frac {3}{q^{17/2}}}-{\frac {2}{q^{19/2}}}+{\frac {1}{q^{21/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{11}z^{-1}+4za^{9}+3a^{9}z^{-1}-z^{5}a^{7}-7z^{3}a^{7}-9za^{7}-3a^{7}z^{-1}+z^{7}a^{5}+6z^{5}a^{5}+10z^{3}a^{5}+8za^{5}+2a^{5}z^{-1}-z^{5}a^{3}-5z^{3}a^{3}-5za^{3}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{12}z^{6}-4a^{12}z^{4}+3a^{12}z^{2}-a^{12}+2a^{11}z^{7}-8a^{11}z^{5}+6a^{11}z^{3}-3a^{11}z+a^{11}z^{-1}+2a^{10}z^{8}-8a^{10}z^{6}+4a^{10}z^{4}+3a^{10}z^{2}-2a^{10}+a^{9}z^{9}-3a^{9}z^{7}-5a^{9}z^{5}+15a^{9}z^{3}-11a^{9}z+3a^{9}z^{-1}+3a^{8}z^{8}-17a^{8}z^{6}+24a^{8}z^{4}-9a^{8}z^{2}+a^{7}z^{9}-4a^{7}z^{7}-5a^{7}z^{5}+22a^{7}z^{3}-14a^{7}z+3a^{7}z^{-1}+2a^{6}z^{8}-14a^{6}z^{6}+25a^{6}z^{4}-14a^{6}z^{2}+2a^{6}+2a^{5}z^{7}-14a^{5}z^{5}+23a^{5}z^{3}-12a^{5}z+2a^{5}z^{-1}+a^{4}z^{8}-6a^{4}z^{6}+9a^{4}z^{4}-5a^{4}z^{2}+a^{3}z^{7}-6a^{3}z^{5}+10a^{3}z^{3}-6a^{3}z+a^{3}z^{-1}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

0

1

-2

0

-4

1

2

1

0

-6

1

-8

2

1

-1

-10

3

2

1

2

-12

2

4

1

-14

2

1

-16

1

2

1

0

-18

1

2

-1

-20

1

-22

1

-1

Integral Khovanov Homology

(db, data source)

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