L11n101

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_13,21,14,20 X_16,7,17,8 X_8,19,9,20 X_18,9,19,10 X_10,17,11,18 X_15,5,16,22 X_21,15,22,14 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -5, 6, -7, 11, -2, -3, 9, -8, -4, 7, -6, 5, 3, -9, 8}

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

Polynomial invariants

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

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

6

1

-1

4

3

2

4

1

-3

0

6

3

-2

6

5

-1

-4

6

5

1

-6

4

6

2

-8

3

6

-3

-10

2

5

3

-12

2

-2

-14

2

Integral Khovanov Homology

(db, data source)

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