L11n93

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_16,8,17,7 X_11,20,12,21 X_17,22,18,5 X_21,18,22,19 X_19,10,20,11 X_14,10,15,9 X_8,16,9,15 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 8, 7, -4, -2, 11, -8, 9, -3, -5, 6, -7, 4, -6, 5}

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

Polynomial invariants

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

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

2

6

3

1

-2

4

5

2

3

2

4

3

-1

0

6

5

1

-2

4

6

2

-4

3

4

-1

-6

1

4

3

-8

3

-3

-10

1

Integral Khovanov Homology

(db, data source)

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