L11n153

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_7,14,8,15 X_13,20,14,21 X_19,1,20,6 X_18,11,19,12 X_5,12,6,13 X_15,22,16,7 X_4,18,5,17 X_21,16,22,17
Gauss code	{1, -2, 3, -10, -8, 6}, {-4, -1, 2, -3, 7, 8, -5, 4, -9, 11, 10, -7, -6, 5, -11, 9}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2t(2)^{2}t(1)^{2}-2t(2)t(1)^{2}-4t(2)^{2}t(1)+7t(2)t(1)-4t(1)-2t(2)+2}{t(1)t(2)}}$ (db)
Jones polynomial	${\frac {7}{q^{9/2}}}-{\frac {6}{q^{7/2}}}+{\frac {3}{q^{5/2}}}-{\frac {2}{q^{3/2}}}+{\frac {1}{q^{21/2}}}-{\frac {2}{q^{19/2}}}+{\frac {4}{q^{17/2}}}-{\frac {6}{q^{15/2}}}+{\frac {7}{q^{13/2}}}-{\frac {8}{q^{11/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}\left(-z^{3}\right)-2a^{9}z-a^{9}z^{-1}+a^{7}z^{5}+3a^{7}z^{3}+4a^{7}z+2a^{7}z^{-1}+a^{5}z^{5}+2a^{5}z^{3}+a^{5}z-2a^{3}z^{3}-4a^{3}z-a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{12}z^{6}-4a^{12}z^{4}+4a^{12}z^{2}+2a^{11}z^{7}-7a^{11}z^{5}+6a^{11}z^{3}-a^{11}z+2a^{10}z^{8}-6a^{10}z^{6}+6a^{10}z^{4}-6a^{10}z^{2}+2a^{10}+a^{9}z^{9}-a^{9}z^{7}-5a^{9}z^{3}+3a^{9}z-a^{9}z^{-1}+4a^{8}z^{8}-13a^{8}z^{6}+20a^{8}z^{4}-18a^{8}z^{2}+5a^{8}+a^{7}z^{9}-a^{7}z^{7}+3a^{7}z^{5}-7a^{7}z^{3}+7a^{7}z-2a^{7}z^{-1}+2a^{6}z^{8}-5a^{6}z^{6}+10a^{6}z^{4}-7a^{6}z^{2}+3a^{6}+2a^{5}z^{7}-4a^{5}z^{5}+7a^{5}z^{3}-2a^{5}z+a^{4}z^{6}+a^{4}z^{2}-a^{4}+3a^{3}z^{3}-5a^{3}z+a^{3}z^{-1}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-2

2

-4

2

1

-1

-6

4

1

3

-8

4

3

-1

-10

4

3

1

-12

3

4

1

-14

3

4

-1

-16

1

3

2

-18

1

3

-2

-20

1

-22

1

-1

Integral Khovanov Homology

(db, data source)

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