L11n155

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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_7,16,8,17 X_13,20,14,21 X_15,22,16,7 X_6,19,1,20 X_18,11,19,12 X_12,6,13,5 X_21,14,22,15 X_4,18,5,17
Gauss code	{1, -2, 3, -11, 9, -7}, {-4, -1, 2, -3, 8, -9, -5, 10, -6, 4, 11, -8, 7, 5, -10, 6}

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

Polynomial invariants

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

-4

2

-6

2

1

-1

-8

5

1

4

-10

4

3

-1

-12

6

4

2

-14

4

0

-16

4

6

-2

-18

2

4

2

-20

1

4

-3

-22

2

-24

1

-1

Integral Khovanov Homology

(db, data source)

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