L11n68

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_7,16,8,17 X_17,22,18,5 X_11,18,12,19 X_13,20,14,21 X_19,12,20,13 X_21,14,22,15 X_15,8,16,9 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, -5, 7, -6, 8, -9, 3, -4, 5, -7, 6, -8, 4}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {uv^{5}-uv^{3}+uv^{2}-uv+u+v^{5}-v^{4}+v^{3}-v^{2}+1}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{7/2}}}-{\frac {1}{q^{13/2}}}+{\frac {1}{q^{15/2}}}-{\frac {3}{q^{17/2}}}+{\frac {2}{q^{19/2}}}-{\frac {2}{q^{21/2}}}+{\frac {2}{q^{23/2}}}-{\frac {1}{q^{25/2}}}+{\frac {1}{q^{27/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-za^{13}-2a^{13}z^{-1}+2z^{3}a^{11}+6za^{11}+4a^{11}z^{-1}+z^{3}a^{9}+za^{9}-a^{9}z^{-1}-z^{7}a^{7}-7z^{5}a^{7}-14z^{3}a^{7}-8za^{7}-a^{7}z^{-1}$ (db)
Kauffman polynomial	$-z^{6}a^{16}+5z^{4}a^{16}-6z^{2}a^{16}+2a^{16}-z^{7}a^{15}+4z^{5}a^{15}-2z^{3}a^{15}-za^{15}-z^{8}a^{14}+4z^{6}a^{14}-2z^{4}a^{14}-2z^{2}a^{14}+a^{14}-z^{9}a^{13}+6z^{7}a^{13}-13z^{5}a^{13}+16z^{3}a^{13}-10za^{13}+2a^{13}z^{-1}-2z^{8}a^{12}+13z^{6}a^{12}-28z^{4}a^{12}+24z^{2}a^{12}-6a^{12}-z^{9}a^{11}+8z^{7}a^{11}-23z^{5}a^{11}+30z^{3}a^{11}-18za^{11}+4a^{11}z^{-1}-z^{8}a^{10}+8z^{6}a^{10}-20z^{4}a^{10}+18z^{2}a^{10}-5a^{10}+z^{5}a^{9}-2z^{3}a^{9}-za^{9}+a^{9}z^{-1}+z^{4}a^{8}-2z^{2}a^{8}+a^{8}-z^{7}a^{7}+7z^{5}a^{7}-14z^{3}a^{7}+8za^{7}-a^{7}z^{-1}$ (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		\

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-6

1

-8

1

-10

1

0

-12

1

-14

2

3

1

0

-16

2

-18

1

3

1

-20

2

0

-22

1

0

-24

1

2

-1

-26

0

-28

1

-1

Integral Khovanov Homology

(db, data source)

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