L11n164

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_10,3,11,4 X_12,17,13,18 X_14,5,15,6 X_4,13,5,14 X_18,11,19,12 X_19,7,20,22 X_15,21,16,20 X_21,17,22,16 X₂₇₃₈ X_6,9,1,10
Gauss code	{1, -10, 2, -5, 4, -11}, {10, -1, 11, -2, 6, -3, 5, -4, -8, 9, 3, -6, -7, 8, -9, 7}

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

Polynomial invariants

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

2

-2

3

2

-1

-4

3

1

2

-6

3

0

-8

3

0

-10

2

3

1

-12

2

3

-1

-14

1

3

2

-16

1

-1

-18

1

Integral Khovanov Homology

(db, data source)

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