L11n454

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Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₃₈₄₉ X_2,16,3,15 X_16,7,17,8 X_19,22,20,15 X_21,14,22,11 X_13,20,14,21 X_9,18,10,19 X_11,10,12,5 X_17,1,18,4
Gauss code	{1, -4, -3, 11}, {-10, 2, -8, 7}, {-2, -1, 5, 3, -9, 10}, {4, -5, -11, 9, -6, 8, -7, 6}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2uvwx^{2}-2uvwx-uvx^{2}+uvx-uwx^{2}+2uwx+ux^{2}-2ux-2vw^{2}x+vw^{2}+2vwx-vw+w^{2}x-w^{2}-2wx+2w}{{\sqrt {u}}{\sqrt {v}}wx}}$ (db)
Jones polynomial	${\frac {6}{q^{9/2}}}-{\frac {7}{q^{7/2}}}+{\frac {2}{q^{5/2}}}-{\frac {2}{q^{3/2}}}+{\frac {1}{q^{21/2}}}-{\frac {3}{q^{19/2}}}+{\frac {4}{q^{17/2}}}-{\frac {7}{q^{15/2}}}+{\frac {7}{q^{13/2}}}-{\frac {9}{q^{11/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}\left(-z^{3}\right)+a^{9}z^{-3}-a^{9}z+a^{9}z^{-1}+a^{7}z^{5}+2a^{7}z^{3}-3a^{7}z^{-3}-a^{7}z-6a^{7}z^{-1}+a^{5}z^{5}+3a^{5}z^{3}+3a^{5}z^{-3}+7a^{5}z+9a^{5}z^{-1}-2a^{3}z^{3}-a^{3}z^{-3}-5a^{3}z-4a^{3}z^{-1}$ (db)
Kauffman polynomial	$-z^{6}a^{12}+3z^{4}a^{12}-z^{2}a^{12}-3z^{7}a^{11}+11z^{5}a^{11}-10z^{3}a^{11}+3za^{11}-3z^{8}a^{10}+9z^{6}a^{10}-4z^{4}a^{10}-z^{2}a^{10}-z^{9}a^{9}-3z^{7}a^{9}+21z^{5}a^{9}-28z^{3}a^{9}+16za^{9}-5a^{9}z^{-1}+a^{9}z^{-3}-5z^{8}a^{8}+15z^{6}a^{8}-10z^{4}a^{8}-7z^{2}a^{8}-3a^{8}z^{-2}+10a^{8}-z^{9}a^{7}-2z^{7}a^{7}+16z^{5}a^{7}-31z^{3}a^{7}+25za^{7}-12a^{7}z^{-1}+3a^{7}z^{-3}-2z^{8}a^{6}+4z^{6}a^{6}-z^{4}a^{6}-15z^{2}a^{6}-6a^{6}z^{-2}+19a^{6}-2z^{7}a^{5}+6z^{5}a^{5}-16z^{3}a^{5}+19za^{5}-12a^{5}z^{-1}+3a^{5}z^{-3}-z^{6}a^{4}+2z^{4}a^{4}-8z^{2}a^{4}-3a^{4}z^{-2}+10a^{4}-3z^{3}a^{3}+7za^{3}-5a^{3}z^{-1}+a^{3}z^{-3}$ (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

χ

-2

2

-4

2

0

-6

5

-8

3

4

1

-10

6

3

-12

3

5

2

-14

4

0

-16

2

5

3

-18

1

2

-1

-20

2

-22

1

-1

Integral Khovanov Homology

(db, data source)

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