L11n16

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Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_4,15,1,16 X_5,12,6,13 X₃₈₄₉ X_9,16,10,17 X_11,20,12,21 X_17,22,18,5 X_21,18,22,19 X_19,10,20,11 X_13,2,14,3
Gauss code	{1, 11, -5, -3}, {-4, -1, 2, 5, -6, 10, -7, 4, -11, -2, 3, 6, -8, 9, -10, 7, -9, 8}

A Braid Representative

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Polynomial invariants

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

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-6

1

-8

1

0

-10

2

-12

1

-14

1

4

2

1

-16

1

2

1

0

-18

3

0

-20

1

2

1

-22

1

2

-1

-24

1

0

-26

1

-1

Integral Khovanov Homology

(db, data source)

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