L11n176

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_20,9,21,10 X_14,5,15,6 X_11,18,12,19 X_3,10,4,11 X_12,7,13,8 X_16,13,17,14 X_22,17,7,18 X_6,15,1,16 X_4,21,5,22 X_19,2,20,3
Gauss code	{1, 11, -5, -10, 3, -9}, {6, -1, 2, 5, -4, -6, 7, -3, 9, -7, 8, 4, -11, -2, 10, -8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2u^{2}v^{4}-u^{2}v^{3}-uv^{4}+uv^{3}+uv^{2}+uv-u-v+2}{uv^{2}}}$ (db)
Jones polynomial	${\frac {1}{q^{9/2}}}-{\frac {1}{q^{7/2}}}-{\frac {1}{q^{23/2}}}+{\frac {2}{q^{21/2}}}-{\frac {2}{q^{19/2}}}+{\frac {3}{q^{17/2}}}-{\frac {3}{q^{15/2}}}+{\frac {2}{q^{13/2}}}-{\frac {3}{q^{11/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$a^{13}(-z)+a^{11}z^{5}+5a^{11}z^{3}+4a^{11}z-a^{11}z^{-1}-a^{9}z^{7}-5a^{9}z^{5}-5a^{9}z^{3}+2a^{9}z+3a^{9}z^{-1}-a^{7}z^{7}-6a^{7}z^{5}-11a^{7}z^{3}-8a^{7}z-2a^{7}z^{-1}$ (db)
Kauffman polynomial	$-z^{7}a^{13}+5z^{5}a^{13}-6z^{3}a^{13}+za^{13}-2z^{8}a^{12}+11z^{6}a^{12}-17z^{4}a^{12}+8z^{2}a^{12}+a^{12}-z^{9}a^{11}+4z^{7}a^{11}-2z^{5}a^{11}-z^{3}a^{11}-a^{11}z^{-1}-3z^{8}a^{10}+15z^{6}a^{10}-18z^{4}a^{10}+3z^{2}a^{10}+3a^{10}-z^{9}a^{9}+4z^{7}a^{9}-z^{5}a^{9}-6z^{3}a^{9}+7za^{9}-3a^{9}z^{-1}-z^{8}a^{8}+4z^{6}a^{8}-z^{4}a^{8}-5z^{2}a^{8}+3a^{8}-z^{7}a^{7}+6z^{5}a^{7}-11z^{3}a^{7}+8za^{7}-2a^{7}z^{-1}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-6

1

-8

1

0

-10

2

-12

1

-14

3

2

1

-16

1

2

1

0

-18

2

3

-1

-20

1

0

-22

1

-1

-24

1

Integral Khovanov Homology

(db, data source)

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