L11n35

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Planar diagram presentation	X₆₁₇₂ X_20,7,21,8 X_4,21,1,22 X_9,14,10,15 X₈₄₉₃ X_12,5,13,6 X_22,13,5,14 X_15,18,16,19 X_11,17,12,16 X_17,11,18,10 X_2,20,3,19
Gauss code	{1, -11, 5, -3}, {6, -1, 2, -5, -4, 10, -9, -6, 7, 4, -8, 9, -10, 8, 11, -2, 3, -7}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-2)(t(2)-1)(2t(2)-1)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {10}{q^{9/2}}}+{\frac {12}{q^{7/2}}}-{\frac {12}{q^{5/2}}}-2q^{3/2}+{\frac {11}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {4}{q^{13/2}}}+{\frac {7}{q^{11/2}}}+4{\sqrt {q}}-{\frac {9}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}(-z)+3a^{5}z^{3}+4a^{5}z+a^{5}z^{-1}-2a^{3}z^{5}-6a^{3}z^{3}-8a^{3}z-3a^{3}z^{-1}+4az^{3}+7az+4az^{-1}-2za^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2a^{5}z^{9}-2a^{3}z^{9}-5a^{6}z^{8}-10a^{4}z^{8}-5a^{2}z^{8}-4a^{7}z^{7}-5a^{5}z^{7}-5a^{3}z^{7}-4az^{7}-a^{8}z^{6}+12a^{6}z^{6}+26a^{4}z^{6}+12a^{2}z^{6}-z^{6}+11a^{7}z^{5}+27a^{5}z^{5}+26a^{3}z^{5}+10az^{5}+2a^{8}z^{4}-4a^{6}z^{4}-17a^{4}z^{4}-13a^{2}z^{4}-2z^{4}-7a^{7}z^{3}-22a^{5}z^{3}-33a^{3}z^{3}-21az^{3}-3z^{3}a^{-1}-a^{8}z^{2}-a^{2}z^{2}+a^{7}z+6a^{5}z+15a^{3}z+15az+5za^{-1}+a^{6}+3a^{4}+3a^{2}+2-a^{5}z^{-1}-3a^{3}z^{-1}-4az^{-1}-2a^{-1}z^{-1}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

4

2

-2

0

7

2

5

-2

6

4

-2

-4

6

5

1

-6

6

0

-8

4

6

-2

-10

3

6

3

-12

1

4

-3

-14

3

-16

1

-1

Integral Khovanov Homology

(db, data source)

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