L11n184

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_9,21,10,20 X_6,21,1,22 X_18,8,19,7 X_3,10,4,11 X_15,12,16,13 X_5,14,6,15 X_13,4,14,5 X_11,16,12,17 X_22,18,7,17 X_19,2,20,3
Gauss code	{1, 11, -5, 8, -7, -3}, {4, -1, -2, 5, -9, 6, -8, 7, -6, 9, 10, -4, -11, 2, 3, -10}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2t(1)t(2)^{4}-t(2)^{4}-2t(1)t(2)^{3}+2t(2)^{3}+t(1)t(2)^{2}+2t(1)^{2}t(2)-2t(1)t(2)-t(1)^{2}+2t(1)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-{\frac {5}{q^{9/2}}}+{\frac {4}{q^{7/2}}}-{\frac {5}{q^{5/2}}}+{\frac {3}{q^{3/2}}}-{\frac {1}{q^{17/2}}}+{\frac {2}{q^{15/2}}}-{\frac {3}{q^{13/2}}}+{\frac {4}{q^{11/2}}}+{\sqrt {q}}-{\frac {2}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^{3}a^{7}+2za^{7}-z^{5}a^{5}-3z^{3}a^{5}-za^{5}+2a^{5}z^{-1}-z^{5}a^{3}-4z^{3}a^{3}-6za^{3}-3a^{3}z^{-1}+z^{3}a+2za+az^{-1}$ (db)
Kauffman polynomial	$a^{9}z^{7}-5a^{9}z^{5}+7a^{9}z^{3}-3a^{9}z+2a^{8}z^{8}-10a^{8}z^{6}+14a^{8}z^{4}-6a^{8}z^{2}+a^{7}z^{9}-2a^{7}z^{7}-6a^{7}z^{5}+10a^{7}z^{3}-2a^{7}z+4a^{6}z^{8}-18a^{6}z^{6}+22a^{6}z^{4}-9a^{6}z^{2}+a^{5}z^{9}-a^{5}z^{7}-9a^{5}z^{5}+14a^{5}z^{3}-7a^{5}z+2a^{5}z^{-1}+2a^{4}z^{8}-7a^{4}z^{6}+5a^{4}z^{4}+2a^{4}z^{2}-3a^{4}+2a^{3}z^{7}-8a^{3}z^{5}+13a^{3}z^{3}-10a^{3}z+3a^{3}z^{-1}+a^{2}z^{6}-3a^{2}z^{4}+6a^{2}z^{2}-3a^{2}+2az^{3}-2az+az^{-1}+z^{2}-1$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

χ

2

1

-1

0

1

-2

3

2

-1

-4

2

-6

2

3

1

-8

3

2

1

-10

1

2

1

-12

2

3

-1

-14

1

2

1

-16

1

-1

-18

1

Integral Khovanov Homology

(db, data source)

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