L11n199

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_12,3,13,4 X_5,14,6,15 X_16,7,17,8 X_20,15,21,16 X_13,18,14,19 X_21,6,22,7 X_17,22,18,9 X_4,19,5,20 X_2,9,3,10 X_8,11,1,12
Gauss code	{1, -10, 2, -9, -3, 7, 4, -11}, {10, -1, 11, -2, -6, 3, 5, -4, -8, 6, 9, -5, -7, 8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{3}v^{4}+u^{2}v^{5}-u^{2}v^{4}+u^{2}v^{2}+uv^{3}-uv+u+v}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{7/2}}}+{\frac {1}{q^{9/2}}}-{\frac {2}{q^{11/2}}}+{\frac {1}{q^{13/2}}}-{\frac {1}{q^{15/2}}}+{\frac {1}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$-za^{13}-a^{13}z^{-1}+z^{5}a^{11}+6z^{3}a^{11}+8za^{11}+3a^{11}z^{-1}-z^{7}a^{9}-6z^{5}a^{9}-10z^{3}a^{9}-6za^{9}-2a^{9}z^{-1}-z^{7}a^{7}-6z^{5}a^{7}-10z^{3}a^{7}-5za^{7}$ (db)
Kauffman polynomial	$z^{2}a^{14}-a^{14}+z^{3}a^{13}-za^{13}+a^{13}z^{-1}-z^{8}a^{12}+7z^{6}a^{12}-15z^{4}a^{12}+13z^{2}a^{12}-3a^{12}-z^{9}a^{11}+7z^{7}a^{11}-17z^{5}a^{11}+22z^{3}a^{11}-14za^{11}+3a^{11}z^{-1}-2z^{8}a^{10}+12z^{6}a^{10}-20z^{4}a^{10}+12z^{2}a^{10}-3a^{10}-z^{9}a^{9}+6z^{7}a^{9}-11z^{5}a^{9}+11z^{3}a^{9}-8za^{9}+2a^{9}z^{-1}-z^{8}a^{8}+5z^{6}a^{8}-5z^{4}a^{8}-z^{7}a^{7}+6z^{5}a^{7}-10z^{3}a^{7}+5za^{7}$ (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

χ

-6

1

-8

1

0

-10

1

-12

1

-14

1

2

1

0

-16

1

0

-18

2

0

-20

1

-22

1

0

Integral Khovanov Homology

(db, data source)

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