L10n102

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_14,7,15,8 X_8,13,5,14 X_11,18,12,19 X_15,20,16,17 X_19,16,20,9 X_17,12,18,13 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -4}, {-8, 5, -7, 6}, {10, -2, -5, 8, 4, -3, -6, 7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(x-1)\left(uvwx+uvw-uw+ux^{2}-vw+vx^{2}-x^{2}-x\right)}{{\sqrt {u}}{\sqrt {v}}{\sqrt {w}}x^{3/2}}}$ (db)
Jones polynomial	${\frac {1}{q^{9/2}}}-{\frac {1}{q^{5/2}}}-{\frac {1}{q^{23/2}}}+{\frac {1}{q^{21/2}}}-{\frac {4}{q^{19/2}}}+{\frac {2}{q^{17/2}}}-{\frac {4}{q^{15/2}}}+{\frac {1}{q^{13/2}}}-{\frac {3}{q^{11/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{13}z^{-3}-4a^{11}z^{-1}-3a^{11}z^{-3}+5za^{9}+7a^{9}z^{-1}+3a^{9}z^{-3}-z^{3}a^{7}-za^{7}-2a^{7}z^{-1}-a^{7}z^{-3}-z^{5}a^{5}-5z^{3}a^{5}-4za^{5}-a^{5}z^{-1}$ (db)
Kauffman polynomial	$-z^{7}a^{13}+6z^{5}a^{13}-13z^{3}a^{13}+13za^{13}-6a^{13}z^{-1}+a^{13}z^{-3}-z^{8}a^{12}+3z^{6}a^{12}+4z^{4}a^{12}-16z^{2}a^{12}-3a^{12}z^{-2}+13a^{12}-5z^{7}a^{11}+26z^{5}a^{11}-41z^{3}a^{11}+31za^{11}-14a^{11}z^{-1}+3a^{11}z^{-3}-z^{8}a^{10}+19z^{4}a^{10}-36z^{2}a^{10}-6a^{10}z^{-2}+24a^{10}-4z^{7}a^{9}+22z^{5}a^{9}-36z^{3}a^{9}+26za^{9}-12a^{9}z^{-1}+3a^{9}z^{-3}-3z^{6}a^{8}+16z^{4}a^{8}-21z^{2}a^{8}-3a^{8}z^{-2}+11a^{8}+z^{5}a^{7}-3z^{3}a^{7}+4za^{7}-3a^{7}z^{-1}+a^{7}z^{-3}+z^{4}a^{6}-z^{2}a^{6}-a^{6}-z^{5}a^{5}+5z^{3}a^{5}-4za^{5}+a^{5}z^{-1}$ (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		\

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-4

1

-6

1

-8

2

1

-1

-10

2

-12

3

6

1

2

-14

1

3

-16

1

3

2

-18

3

1

2

-20

1

4

3

-22

0

-24

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-6$	$i=-4$	$i=-2$
$r=-10$		${\mathbb {Z} }$
$r=-9$		${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-8$		${\mathbb {Z} }^{4}$	${\mathbb {Z} }^{3}$
$r=-7$		${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$		${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$		${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-4$	${\mathbb {Z} }^{3}$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{2}$
$r=-3$	${\mathbb {Z} }$	${\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-2$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$