L11n238

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Planar diagram presentation	X_12,1,13,2 X_7,16,8,17 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_17,8,18,9
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 $u$ , $v$ , $w$ , ...)	${\frac {u^{3}v^{3}-2u^{3}v^{2}-2u^{2}v^{3}+6u^{2}v^{2}-6u^{2}v+2u^{2}+2uv^{3}-6uv^{2}+6uv-2u-2v+1}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-3q^{7/2}+6q^{5/2}-10q^{3/2}+12{\sqrt {q}}-{\frac {13}{\sqrt {q}}}+{\frac {12}{q^{3/2}}}-{\frac {10}{q^{5/2}}}+{\frac {6}{q^{7/2}}}-{\frac {3}{q^{9/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-a^{3}z^{5}+5az^{5}-2z^{5}a^{-1}-4a^{3}z^{3}+11az^{3}-7z^{3}a^{-1}+z^{3}a^{-3}+a^{5}z-8a^{3}z+12az-9za^{-1}+2za^{-3}+2a^{5}z^{-1}-5a^{3}z^{-1}+6az^{-1}-4a^{-1}z^{-1}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-az^{9}-z^{9}a^{-1}-5a^{2}z^{8}-3z^{8}a^{-2}-8z^{8}-7a^{3}z^{7}-14az^{7}-10z^{7}a^{-1}-3z^{7}a^{-3}-3a^{4}z^{6}+5a^{2}z^{6}+3z^{6}a^{-2}-z^{6}a^{-4}+12z^{6}+17a^{3}z^{5}+44az^{5}+36z^{5}a^{-1}+9z^{5}a^{-3}+9z^{4}a^{-2}+3z^{4}a^{-4}+6z^{4}-6a^{5}z^{3}-29a^{3}z^{3}-51az^{3}-37z^{3}a^{-1}-9z^{3}a^{-3}-a^{4}z^{2}-3a^{2}z^{2}-11z^{2}a^{-2}-3z^{2}a^{-4}-10z^{2}+7a^{5}z+21a^{3}z+29az+19za^{-1}+4za^{-3}+a^{4}+a^{2}+3a^{-2}+a^{-4}+3-2a^{5}z^{-1}-5a^{3}z^{-1}-6az^{-1}-4a^{-1}z^{-1}-a^{-3}z^{-1}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

2

6

4

1

-3

4

6

2

4

2

6

4

-2

0

7

6

1

-2

6

7

1

-4

4

6

-2

-6

2

6

4

-8

1

4

-3

-10

3

Integral Khovanov Homology

(db, data source)

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