L10n38

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_16,13,17,14 X_14,9,15,10 X_10,15,11,16 X_17,5,18,20 X_7,19,8,18 X_19,9,20,8 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -9, 2, -10}, {9, -1, -7, 8, 4, -5, 10, -2, 3, -4, 5, -3, -6, 7, -8, 6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)+t(2))\left(t(2)^{2}-t(2)+1\right)^{2}}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {6}{q^{9/2}}}+{\frac {5}{q^{7/2}}}-{\frac {7}{q^{5/2}}}+{\frac {5}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {2}{q^{13/2}}}+{\frac {4}{q^{11/2}}}+2{\sqrt {q}}-{\frac {4}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}-2a^{7}z^{-1}+3z^{3}a^{5}+9za^{5}+7a^{5}z^{-1}-2z^{5}a^{3}-9z^{3}a^{3}-14za^{3}-7a^{3}z^{-1}+2z^{3}a+4za+2az^{-1}$ (db)
Kauffman polynomial	$-z^{6}a^{8}+4z^{4}a^{8}-5z^{2}a^{8}+2a^{8}-2z^{7}a^{7}+7z^{5}a^{7}-7z^{3}a^{7}+4za^{7}-2a^{7}z^{-1}-z^{8}a^{6}-2z^{6}a^{6}+17z^{4}a^{6}-20z^{2}a^{6}+8a^{6}-6z^{7}a^{5}+20z^{5}a^{5}-22z^{3}a^{5}+17za^{5}-7a^{5}z^{-1}-z^{8}a^{4}-5z^{6}a^{4}+24z^{4}a^{4}-29z^{2}a^{4}+13a^{4}-4z^{7}a^{3}+12z^{5}a^{3}-18z^{3}a^{3}+16za^{3}-7a^{3}z^{-1}-4z^{6}a^{2}+11z^{4}a^{2}-17z^{2}a^{2}+8a^{2}-z^{5}a-3z^{3}a+3za-2az^{-1}-3z^{2}+2$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

χ

2

-2

0

2

-2

4

3

-1

-4

3

1

2

-6

2

4

2

-8

4

3

1

-10

1

3

2

-12

1

3

-2

-14

1

-16

1

-1

Integral Khovanov Homology

(db, data source)

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