L11n235

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_5,20,6,21 X_14,3,15,4 X_4,15,5,16 X_19,22,20,9 X_16,7,17,8 X_18,12,19,11 X_12,18,13,17 X_2,9,3,10 X_8,13,1,14 X_21,6,22,7
Gauss code	{1, -9, 3, -4, -2, 11, 6, -10}, {9, -1, 7, -8, 10, -3, 4, -6, 8, -7, -5, 2, -11, 5}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(2)^{3}t(1)^{3}-2t(2)^{2}t(1)^{3}-t(2)^{3}t(1)^{2}+t(2)^{2}t(1)^{2}-t(1)^{2}-t(2)^{3}t(1)+t(2)t(1)-t(1)-2t(2)+1}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {3}{q^{9/2}}}+{\frac {3}{q^{7/2}}}-{\frac {3}{q^{5/2}}}+{\frac {2}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {1}{q^{15/2}}}-{\frac {2}{q^{13/2}}}+{\frac {2}{q^{11/2}}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$2za^{9}+a^{9}z^{-1}-z^{5}a^{7}-5z^{3}a^{7}-5za^{7}-a^{7}z^{-1}+z^{7}a^{5}+5z^{5}a^{5}+6z^{3}a^{5}+2za^{5}-z^{5}a^{3}-4z^{3}a^{3}-3za^{3}$ (db)
Kauffman polynomial	$-z^{5}a^{11}+5z^{3}a^{11}-4za^{11}+z^{4}a^{10}-z^{2}a^{10}+z^{5}a^{9}-2z^{3}a^{9}+3za^{9}-a^{9}z^{-1}-z^{8}a^{8}+5z^{6}a^{8}-4z^{4}a^{8}-z^{2}a^{8}+a^{8}-z^{9}a^{7}+4z^{7}a^{7}-7z^{3}a^{7}+5za^{7}-a^{7}z^{-1}-3z^{8}a^{6}+15z^{6}a^{6}-18z^{4}a^{6}+5z^{2}a^{6}-z^{9}a^{5}+3z^{7}a^{5}+3z^{5}a^{5}-7z^{3}a^{5}+za^{5}-2z^{8}a^{4}+10z^{6}a^{4}-13z^{4}a^{4}+5z^{2}a^{4}-z^{7}a^{3}+5z^{5}a^{3}-7z^{3}a^{3}+3za^{3}$ (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

1

-1

-4

2

1

-6

2

0

-8

1

2

1

0

-10

1

2

1

-12

1

3

2

0

-14

1

-16

1

2

-1

-18

1

-20

1

Integral Khovanov Homology

(db, data source)

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