L11n294

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_20,16,21,15 X_14,8,15,7 X_10,22,5,21 X_11,19,12,18 X_9,17,10,16 X_17,11,18,22 X_19,9,20,8 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 4, 9, -7, -5}, {-6, -2, 11, -4, 3, 7, -8, 6, -9, -3, 5, 8}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)t(2)^{2}t(3)^{4}-t(2)^{2}t(3)^{4}-t(1)t(2)t(3)^{4}+t(2)^{2}t(3)^{3}+t(1)t(2)t(3)^{2}-t(2)t(3)^{2}-t(1)t(3)+t(1)+t(2)-1}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$-q^{8}+q^{7}-q^{6}+3q^{5}-q^{4}+3q^{3}-q^{2}+q$ (db)
Signature	6 (db)
HOMFLY-PT polynomial	$-z^{8}a^{-6}+z^{6}a^{-4}-7z^{6}a^{-6}+z^{6}a^{-8}+6z^{4}a^{-4}-17z^{4}a^{-6}+6z^{4}a^{-8}+11z^{2}a^{-4}-21z^{2}a^{-6}+10z^{2}a^{-8}-z^{2}a^{-10}+8a^{-4}-15a^{-6}+8a^{-8}-a^{-10}+2a^{-4}z^{-2}-5a^{-6}z^{-2}+4a^{-8}z^{-2}-a^{-10}z^{-2}$ (db)
Kauffman polynomial	$-2za^{-11}+a^{-11}z^{-1}-3z^{2}a^{-10}-a^{-10}z^{-2}+2a^{-10}+2z^{7}a^{-9}-12z^{5}a^{-9}+19z^{3}a^{-9}-13za^{-9}+5a^{-9}z^{-1}+3z^{8}a^{-8}-19z^{6}a^{-8}+36z^{4}a^{-8}-29z^{2}a^{-8}-4a^{-8}z^{-2}+13a^{-8}+z^{9}a^{-7}-3z^{7}a^{-7}-9z^{5}a^{-7}+31z^{3}a^{-7}-27za^{-7}+9a^{-7}z^{-1}+4z^{8}a^{-6}-26z^{6}a^{-6}+53z^{4}a^{-6}-45z^{2}a^{-6}-5a^{-6}z^{-2}+20a^{-6}+z^{9}a^{-5}-5z^{7}a^{-5}+3z^{5}a^{-5}+12z^{3}a^{-5}-16za^{-5}+5a^{-5}z^{-1}+z^{8}a^{-4}-7z^{6}a^{-4}+17z^{4}a^{-4}-19z^{2}a^{-4}-2a^{-4}z^{-2}+10a^{-4}$ (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		\

-2

-1

0

1

2

3

4

5

6

χ

17

2

1

-1

15

1

2

1

0

13

2

0

11

1

2

9

1

3

2

7

2

1

2

5

1

3

2

3

0

1

Integral Khovanov Homology

(db, data source)

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