L11n47

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Planar diagram presentation	X₆₁₇₂ X_18,7,19,8 X_4,19,1,20 X_5,14,6,15 X₃₈₄₉ X_9,16,10,17 X_15,10,16,11 X_11,20,12,21 X_13,22,14,5 X_21,12,22,13 X_17,2,18,3
Gauss code	{1, 11, -5, -3}, {-4, -1, 2, 5, -6, 7, -8, 10, -9, 4, -7, 6, -11, -2, 3, 8, -10, 9}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2uv^{5}-2uv^{4}+2uv^{3}-2uv^{2}+uv+v^{4}-2v^{3}+2v^{2}-2v+2}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	${\frac {1}{q^{9/2}}}-{\frac {1}{q^{7/2}}}+{\frac {2}{q^{25/2}}}-{\frac {3}{q^{23/2}}}+{\frac {5}{q^{21/2}}}-{\frac {6}{q^{19/2}}}+{\frac {5}{q^{17/2}}}-{\frac {6}{q^{15/2}}}+{\frac {4}{q^{13/2}}}-{\frac {3}{q^{11/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$a^{13}(-z)-2a^{13}z^{-1}+a^{11}z^{5}+5a^{11}z^{3}+8a^{11}z+4a^{11}z^{-1}-a^{9}z^{7}-5a^{9}z^{5}-7a^{9}z^{3}-3a^{9}z-a^{9}z^{-1}-a^{7}z^{7}-6a^{7}z^{5}-11a^{7}z^{3}-6a^{7}z-a^{7}z^{-1}$ (db)
Kauffman polynomial	$-3z^{2}a^{16}+2a^{16}-z^{5}a^{15}-z^{3}a^{15}-za^{15}-2z^{6}a^{14}+2z^{4}a^{14}-3z^{2}a^{14}+a^{14}-2z^{7}a^{13}+z^{5}a^{13}+6z^{3}a^{13}-8za^{13}+2a^{13}z^{-1}-2z^{8}a^{12}+6z^{6}a^{12}-12z^{4}a^{12}+17z^{2}a^{12}-6a^{12}-z^{9}a^{11}+3z^{7}a^{11}-9z^{5}a^{11}+22z^{3}a^{11}-16za^{11}+4a^{11}z^{-1}-3z^{8}a^{10}+12z^{6}a^{10}-17z^{4}a^{10}+15z^{2}a^{10}-5a^{10}-z^{9}a^{9}+4z^{7}a^{9}-5z^{5}a^{9}+4z^{3}a^{9}-3za^{9}+a^{9}z^{-1}-z^{8}a^{8}+4z^{6}a^{8}-3z^{4}a^{8}-2z^{2}a^{8}+a^{8}-z^{7}a^{7}+6z^{5}a^{7}-11z^{3}a^{7}+6za^{7}-a^{7}z^{-1}$ (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

2

-12

2

1

-1

-14

4

2

-16

2

3

1

-18

4

3

1

-20

1

2

1

-22

2

4

-2

-24

1

-26

2

-2

Integral Khovanov Homology

(db, data source)

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