L11n132

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_18,11,19,12 X_3,10,4,11 X_17,3,18,2 X_12,5,13,6 X₆₇₁₈ X_9,16,10,17 X_13,20,14,21 X_15,22,16,7 X_4,20,5,19 X_21,14,22,15
Gauss code	{1, 4, -3, -10, 5, -6}, {6, -1, -7, 3, 2, -5, -8, 11, -9, 7, -4, -2, 10, 8, -11, 9}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)t(2)^{4}-t(1)^{2}t(2)^{3}+t(1)^{2}t(2)^{2}-t(1)t(2)^{2}+t(2)^{2}-t(2)+t(1)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{7/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{21/2}}}-{\frac {1}{q^{19/2}}}+{\frac {1}{q^{17/2}}}-{\frac {1}{q^{15/2}}}+{\frac {1}{q^{13/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-z^{3}a^{9}-3za^{9}-2a^{9}z^{-1}+z^{5}a^{7}+6z^{3}a^{7}+11za^{7}+5a^{7}z^{-1}-z^{5}a^{5}-6z^{3}a^{5}-9za^{5}-3a^{5}z^{-1}$ (db)
Kauffman polynomial	$-z^{6}a^{12}+5z^{4}a^{12}-6z^{2}a^{12}+a^{12}-z^{7}a^{11}+5z^{5}a^{11}-6z^{3}a^{11}+za^{11}-z^{6}a^{10}+4z^{4}a^{10}-3z^{2}a^{10}-z^{5}a^{9}+4z^{3}a^{9}-4za^{9}+2a^{9}z^{-1}+z^{6}a^{8}-7z^{4}a^{8}+13z^{2}a^{8}-5a^{8}+z^{7}a^{7}-7z^{5}a^{7}+16z^{3}a^{7}-15za^{7}+5a^{7}z^{-1}+z^{6}a^{6}-6z^{4}a^{6}+10z^{2}a^{6}-5a^{6}-z^{5}a^{5}+6z^{3}a^{5}-10za^{5}+3a^{5}z^{-1}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-4

1

-6

1

2

-8

1

2

1

-10

1

-12

1

2

1

0

-14

1

0

-16

1

0

-18

1

0

-20

0

-22

1

-1

Integral Khovanov Homology

(db, data source)

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