L11n19

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_5,12,6,13 X₃₈₄₉ X_13,22,14,5 X_21,14,22,15 X_9,18,10,19 X_11,20,12,21 X_19,10,20,11 X_15,2,16,3
Gauss code	{1, 11, -5, -3}, {-4, -1, 2, 5, -8, 10, -9, 4, -6, 7, -11, -2, 3, 8, -10, 9, -7, 6}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2uv^{5}-uv^{4}-v+2}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	${\frac {1}{q^{9/2}}}-{\frac {1}{q^{7/2}}}+{\frac {1}{q^{25/2}}}+{\frac {1}{q^{21/2}}}-{\frac {2}{q^{19/2}}}+{\frac {1}{q^{17/2}}}-{\frac {2}{q^{15/2}}}+{\frac {1}{q^{13/2}}}-{\frac {2}{q^{11/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$-za^{13}-2a^{13}z^{-1}+z^{5}a^{11}+6z^{3}a^{11}+9za^{11}+4a^{11}z^{-1}-z^{7}a^{9}-6z^{5}a^{9}-10z^{3}a^{9}-5za^{9}-a^{9}z^{-1}-z^{7}a^{7}-6z^{5}a^{7}-10z^{3}a^{7}-5za^{7}-a^{7}z^{-1}$ (db)
Kauffman polynomial	$-z^{2}a^{16}+2a^{16}-za^{15}-z^{2}a^{14}+a^{14}-z^{5}a^{13}+5z^{3}a^{13}-7za^{13}+2a^{13}z^{-1}-z^{8}a^{12}+7z^{6}a^{12}-17z^{4}a^{12}+18z^{2}a^{12}-6a^{12}-z^{9}a^{11}+7z^{7}a^{11}-18z^{5}a^{11}+24z^{3}a^{11}-15za^{11}+4a^{11}z^{-1}-2z^{8}a^{10}+12z^{6}a^{10}-22z^{4}a^{10}+17z^{2}a^{10}-5a^{10}-z^{9}a^{9}+6z^{7}a^{9}-11z^{5}a^{9}+9z^{3}a^{9}-4za^{9}+a^{9}z^{-1}-z^{8}a^{8}+5z^{6}a^{8}-5z^{4}a^{8}-z^{2}a^{8}+a^{8}-z^{7}a^{7}+6z^{5}a^{7}-10z^{3}a^{7}+5za^{7}-a^{7}z^{-1}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-6

1

-8

1

0

-10

1

-12

1

-14

1

3

1

-16

1

-18

3

2

1

-20

1

-22

1

2

-1

-24

1

-1

-26

1

-1

Integral Khovanov Homology

(db, data source)

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