L11n240

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Planar diagram presentation	X_12,1,13,2 X_16,8,17,7 X_5,1,6,10 X₃₇₄₆ X_9,5,10,4 X_13,18,14,19 X_19,22,20,11 X_15,21,16,20 X_21,15,22,14 X_2,11,3,12 X_8,18,9,17
Gauss code	{1, -10, -4, 5, -3, 4, 2, -11, -5, 3}, {10, -1, -6, 9, -8, -2, 11, 6, -7, 8, -9, 7}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(t(2)t(1)^{2}-t(2)^{2}t(1)+2t(2)t(1)-t(1)+t(2)\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{15/2}-2q^{13/2}+3q^{11/2}-4q^{9/2}+3q^{7/2}-3q^{5/2}+2q^{3/2}-{\sqrt {q}}-{\frac {1}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$za^{-7}+a^{-7}z^{-1}-2z^{3}a^{-5}-5za^{-5}-4a^{-5}z^{-1}+z^{5}a^{-3}+5z^{3}a^{-3}+10za^{-3}+6a^{-3}z^{-1}-z^{5}a^{-1}+az^{3}-6z^{3}a^{-1}+3az-9za^{-1}+2az^{-1}-5a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-8}-4z^{4}a^{-8}+4z^{2}a^{-8}-a^{-8}+2z^{7}a^{-7}-8z^{5}a^{-7}+8z^{3}a^{-7}-4za^{-7}+a^{-7}z^{-1}+z^{8}a^{-6}-z^{6}a^{-6}-8z^{4}a^{-6}+9z^{2}a^{-6}-3a^{-6}+4z^{7}a^{-5}-18z^{5}a^{-5}+24z^{3}a^{-5}-16za^{-5}+4a^{-5}z^{-1}+z^{8}a^{-4}-3z^{6}a^{-4}-2z^{4}a^{-4}+7z^{2}a^{-4}-3a^{-4}+3z^{7}a^{-3}-20z^{5}a^{-3}+39z^{3}a^{-3}-27za^{-3}+6a^{-3}z^{-1}+z^{8}a^{-2}-7z^{6}a^{-2}+10z^{4}a^{-2}-a^{-2}+az^{7}+2z^{7}a^{-1}-6az^{5}-16z^{5}a^{-1}+10az^{3}+33z^{3}a^{-1}-7az-22za^{-1}+2az^{-1}+5a^{-1}z^{-1}+z^{8}-6z^{6}+8z^{4}-2z^{2}-1$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

16

1

-1

14

1

12

2

1

-1

10

2

1

8

1

2

1

6

2

0

4

1

2

1

2

1

2

-1

0

1

3

2

-2

1

0

-4

0

-6

1

Integral Khovanov Homology

(db, data source)

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