L11n62

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

A Braid Representative

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Polynomial invariants

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

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-6

1

-8

1

-10

1

0

-12

3

-14

3

1

-1

-16

3

2

1

-18

2

3

1

0

-20

3

0

-22

1

2

1

-24

1

3

-2

-26

1

-28

1

-1

Integral Khovanov Homology

(db, data source)

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