L11n165

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Planar diagram presentation	X₈₁₉₂ X_10,3,11,4 X_15,20,16,21 X_14,5,15,6 X_4,13,5,14 X_17,22,18,7 X_21,16,22,17 X_19,12,20,13 X_11,18,12,19 X₂₇₃₈ X_6,9,1,10
Gauss code	{1, -10, 2, -5, 4, -11}, {10, -1, 11, -2, -9, 8, 5, -4, -3, 7, -6, 9, -8, 3, -7, 6}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{2}v^{5}-u^{2}v^{3}+u^{2}v^{2}+uv^{6}-uv^{5}+uv^{3}-uv+u+v^{4}-v^{3}+v}{uv^{3}}}$ (db)
Jones polynomial	${\frac {1}{q^{9/2}}}-{\frac {1}{q^{7/2}}}+{\frac {1}{q^{27/2}}}-{\frac {1}{q^{25/2}}}+{\frac {1}{q^{23/2}}}-{\frac {1}{q^{21/2}}}-{\frac {1}{q^{15/2}}}+{\frac {1}{q^{13/2}}}-{\frac {2}{q^{11/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$-2za^{13}-2a^{13}z^{-1}+z^{5}a^{11}+7z^{3}a^{11}+11za^{11}+5a^{11}z^{-1}-z^{7}a^{9}-6z^{5}a^{9}-10z^{3}a^{9}-7za^{9}-3a^{9}z^{-1}-z^{7}a^{7}-6z^{5}a^{7}-10z^{3}a^{7}-5za^{7}$ (db)
Kauffman polynomial	$a^{16}z^{6}-5a^{16}z^{4}+6a^{16}z^{2}-a^{16}+a^{15}z^{7}-5a^{15}z^{5}+6a^{15}z^{3}-2a^{15}z+a^{14}z^{6}-5a^{14}z^{4}+4a^{14}z^{2}-3a^{13}z^{3}+4a^{13}z-2a^{13}z^{-1}+a^{12}z^{8}-8a^{12}z^{6}+19a^{12}z^{4}-19a^{12}z^{2}+5a^{12}+a^{11}z^{9}-8a^{11}z^{7}+22a^{11}z^{5}-31a^{11}z^{3}+21a^{11}z-5a^{11}z^{-1}+2a^{10}z^{8}-13a^{10}z^{6}+24a^{10}z^{4}-17a^{10}z^{2}+5a^{10}+a^{9}z^{9}-6a^{9}z^{7}+11a^{9}z^{5}-12a^{9}z^{3}+10a^{9}z-3a^{9}z^{-1}+a^{8}z^{8}-5a^{8}z^{6}+5a^{8}z^{4}+a^{7}z^{7}-6a^{7}z^{5}+10a^{7}z^{3}-5a^{7}z$ (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		\

-11

-10

-9

-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

-18

1

3

2

0

-20

1

-22

1

0

-24

1

0

-26

0

-28

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-8$	$i=-6$	$i=-4$
$r=-11$		${\mathbb {Z} }$
$r=-10$		${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-9$		${\mathbb {Z} }$
$r=-8$		${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }^{3}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{2}$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$