L11n84

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_14,8,15,7 X_9,18,10,19 X_19,5,20,22 X_15,21,16,20 X_21,17,22,16 X_17,8,18,9 X_10,14,11,13 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, 8, -4, -9, 11, -2, 9, -3, -6, 7, -8, 4, -5, 6, -7, 5}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)t(2)^{3}-t(2)^{3}-3t(1)t(2)^{2}+4t(2)^{2}+4t(1)t(2)-3t(2)-t(1)+1}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-3q^{7/2}+4q^{5/2}-5q^{3/2}+6{\sqrt {q}}-{\frac {6}{\sqrt {q}}}+{\frac {5}{q^{3/2}}}-{\frac {4}{q^{5/2}}}+{\frac {1}{q^{7/2}}}-{\frac {1}{q^{9/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{-1}-2za^{3}-a^{3}z^{-1}+2z^{3}a+2za-z^{5}a^{-1}-3z^{3}a^{-1}-3za^{-1}+z^{3}a^{-3}+za^{-3}$ (db)
Kauffman polynomial	$a^{5}z^{3}-2a^{5}z+a^{5}z^{-1}+z^{6}a^{-4}+a^{4}z^{4}-3z^{4}a^{-4}+z^{2}a^{-4}-a^{4}+3z^{7}a^{-3}+2a^{3}z^{5}-11z^{5}a^{-3}-2a^{3}z^{3}+8z^{3}a^{-3}-za^{-3}+a^{3}z^{-1}+a^{2}z^{8}+3z^{8}a^{-2}-4a^{2}z^{6}-11z^{6}a^{-2}+9a^{2}z^{4}+9z^{4}a^{-2}-5a^{2}z^{2}-2z^{2}a^{-2}+az^{9}+z^{9}a^{-1}-3az^{7}+3az^{5}-10z^{5}a^{-1}+11z^{3}a^{-1}+az-2za^{-1}+4z^{8}-16z^{6}+20z^{4}-8z^{2}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

2

6

2

1

-1

4

3

2

1

2

3

2

-1

0

3

0

-2

3

4

1

-4

1

2

-1

-6

3

-8

1

0

-10

1

Integral Khovanov Homology

(db, data source)

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