L11n161

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_16,7,17,8 X_10,4,11,3 X_2,15,3,16 X_14,10,15,9 X_18,11,19,12 X_5,13,6,12 X_6,21,1,22 X_13,20,14,21 X_22,17,7,18 X_19,4,20,5
Gauss code	{1, -4, 3, 11, -7, -8}, {2, -1, 5, -3, 6, 7, -9, -5, 4, -2, 10, -6, -11, 9, 8, -10}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2t(2)^{2}t(1)^{2}-3t(2)t(1)^{2}+t(1)^{2}-2t(2)^{2}t(1)+5t(2)t(1)-2t(1)+t(2)^{2}-3t(2)+2}{t(1)t(2)}}$ (db)
Jones polynomial	$-{\frac {7}{q^{9/2}}}+{\frac {6}{q^{7/2}}}-{\frac {7}{q^{5/2}}}+{\frac {4}{q^{3/2}}}-{\frac {1}{q^{17/2}}}+{\frac {3}{q^{15/2}}}-{\frac {4}{q^{13/2}}}+{\frac {6}{q^{11/2}}}+{\sqrt {q}}-{\frac {3}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}z^{3}+a^{7}z-a^{7}z^{-1}-a^{5}z^{5}-2a^{5}z^{3}+a^{5}z+3a^{5}z^{-1}-a^{3}z^{5}-3a^{3}z^{3}-4a^{3}z-2a^{3}z^{-1}+az^{3}+az$ (db)
Kauffman polynomial	$a^{9}z^{7}-4a^{9}z^{5}+4a^{9}z^{3}-a^{9}z+3a^{8}z^{8}-14a^{8}z^{6}+18a^{8}z^{4}-6a^{8}z^{2}-a^{8}+2a^{7}z^{9}-6a^{7}z^{7}-a^{7}z^{5}+7a^{7}z^{3}-2a^{7}z+a^{7}z^{-1}+7a^{6}z^{8}-30a^{6}z^{6}+34a^{6}z^{4}-9a^{6}z^{2}-3a^{6}+2a^{5}z^{9}-4a^{5}z^{7}-6a^{5}z^{5}+10a^{5}z^{3}-5a^{5}z+3a^{5}z^{-1}+4a^{4}z^{8}-15a^{4}z^{6}+16a^{4}z^{4}-4a^{4}z^{2}-3a^{4}+3a^{3}z^{7}-9a^{3}z^{5}+10a^{3}z^{3}-6a^{3}z+2a^{3}z^{-1}+a^{2}z^{6}+3az^{3}-2az+z^{2}$ (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

4

1

3

-6

3

4

1

-8

4

3

1

-10

2

3

1

-12

2

4

-2

-14

1

2

1

-16

2

-2

-18

1

Integral Khovanov Homology

(db, data source)

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