L10n108

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₃₈₄₉ X_15,2,16,3 X_16,7,17,8 X_19,11,20,14 X_13,15,14,20 X_9,18,10,19 X_11,10,12,5 X_4,17,1,18
Gauss code	{1, 4, -3, -10}, {-9, 2, -7, 6}, {-2, -1, 5, 3, -8, 9}, {-4, -5, 10, 8, -6, 7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uvw^{2}x^{2}-uvw^{2}x-uw^{2}x^{2}+uwx^{2}+vw-v-x+1}{{\sqrt {u}}{\sqrt {v}}wx}}$ (db)
Jones polynomial	$-{\frac {1}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}-{\frac {3}{q^{5/2}}}+{\frac {1}{q^{7/2}}}-{\frac {4}{q^{9/2}}}+{\frac {1}{q^{11/2}}}-{\frac {3}{q^{13/2}}}+{\frac {1}{q^{15/2}}}-{\frac {1}{q^{17/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$za^{9}+a^{9}z^{-1}+a^{9}z^{-3}-z^{5}a^{7}-5z^{3}a^{7}-7za^{7}-6a^{7}z^{-1}-3a^{7}z^{-3}+z^{7}a^{5}+6z^{5}a^{5}+12z^{3}a^{5}+13za^{5}+9a^{5}z^{-1}+3a^{5}z^{-3}-z^{5}a^{3}-5z^{3}a^{3}-7za^{3}-4a^{3}z^{-1}-a^{3}z^{-3}$ (db)
Kauffman polynomial	$a^{11}z+a^{10}z^{2}+3a^{9}z^{3}-a^{9}z^{-3}-8a^{9}z+5a^{9}z^{-1}+2a^{8}z^{6}-9a^{8}z^{4}+13a^{8}z^{2}+3a^{8}z^{-2}-10a^{8}+3a^{7}z^{7}-16a^{7}z^{5}+28a^{7}z^{3}-3a^{7}z^{-3}-25a^{7}z+12a^{7}z^{-1}+a^{6}z^{8}-2a^{6}z^{6}-9a^{6}z^{4}+23a^{6}z^{2}+6a^{6}z^{-2}-19a^{6}+4a^{5}z^{7}-22a^{5}z^{5}+37a^{5}z^{3}-3a^{5}z^{-3}-27a^{5}z+12a^{5}z^{-1}+a^{4}z^{8}-4a^{4}z^{6}+11a^{4}z^{2}+3a^{4}z^{-2}-10a^{4}+a^{3}z^{7}-6a^{3}z^{5}+12a^{3}z^{3}-a^{3}z^{-3}-11a^{3}z+5a^{3}z^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

χ

0

1

-2

0

-4

3

1

2

-6

1

2

-8

4

1

3

-10

2

1

2

3

-12

4

2

-14

1

2

-16

2

0

-18

1

Integral Khovanov Homology

(db, data source)

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