L11n332

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Planar diagram presentation	X₆₁₇₂ X_11,16,12,17 X₃₈₄₉ X_17,2,18,3 X_5,14,6,15 X_18,7,19,8 X_15,12,16,5 X_13,20,14,21 X_9,13,10,22 X_21,11,22,10 X_4,19,1,20
Gauss code	{1, 4, -3, -11}, {-5, -1, 6, 3, -9, 10, -2, 7}, {-8, 5, -7, 2, -4, -6, 11, 8, -10, 9}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(v+w-1)(vw-v-w)(uvw-1)}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$-q^{-9}+2q^{-8}-3q^{-7}+7q^{-6}-5q^{-5}+6q^{-4}-5q^{-3}+4q^{-2}-2q^{-1}+1$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{10}+z^{4}a^{8}+4z^{2}a^{8}+a^{8}z^{-2}+2a^{8}-z^{6}a^{6}-4z^{4}a^{6}-4z^{2}a^{6}-2a^{6}z^{-2}-3a^{6}-z^{6}a^{4}-3z^{4}a^{4}+a^{4}z^{-2}+a^{4}+z^{4}a^{2}+3z^{2}a^{2}+a^{2}$ (db)
Kauffman polynomial	$z^{3}a^{11}-2za^{11}+2z^{4}a^{10}-4z^{2}a^{10}+3a^{10}+z^{7}a^{9}-4z^{5}a^{9}+10z^{3}a^{9}-7za^{9}+2z^{8}a^{8}-10z^{6}a^{8}+22z^{4}a^{8}-19z^{2}a^{8}-a^{8}z^{-2}+11a^{8}+z^{9}a^{7}-2z^{7}a^{7}-3z^{5}a^{7}+14z^{3}a^{7}-12za^{7}+2a^{7}z^{-1}+4z^{8}a^{6}-17z^{6}a^{6}+27z^{4}a^{6}-22z^{2}a^{6}-2a^{6}z^{-2}+11a^{6}+z^{9}a^{5}-z^{7}a^{5}-6z^{5}a^{5}+10z^{3}a^{5}-8za^{5}+2a^{5}z^{-1}+2z^{8}a^{4}-6z^{6}a^{4}+3z^{4}a^{4}-3z^{2}a^{4}-a^{4}z^{-2}+3a^{4}+2z^{7}a^{3}-7z^{5}a^{3}+5z^{3}a^{3}-za^{3}+z^{6}a^{2}-4z^{4}a^{2}+4z^{2}a^{2}-a^{2}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

1

-1

1

-1

-3

3

1

2

-5

3

2

-1

-7

3

2

1

-9

1

3

1

-11

5

3

2

-13

1

5

4

-15

1

2

-1

-17

1

-19

1

-1

Integral Khovanov Homology

(db, data source)

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