L11n237

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_3,14,4,15 X_22,5,9,6 X_6,9,7,10 X_20,12,21,11 X_18,14,19,13 X_12,20,13,19 X_7,16,8,17 X_15,4,16,5 X_17,8,18,1 X_2,21,3,22
Gauss code	{1, -11, -2, 9, 3, -4, -8, 10}, {4, -1, 5, -7, 6, 2, -9, 8, -10, -6, 7, -5, 11, -3}

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

Polynomial invariants

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

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

0

1

-2

0

-4

1

2

1

0

-6

1

0

-8

2

1

-1

-10

1

-12

1

3

1

-14

1

-16

1

0

-18

1

0

-20

1

Integral Khovanov Homology

(db, data source)

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