L11a186

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Planar diagram presentation	X₈₁₉₂ X_10,3,11,4 X_22,18,7,17 X_16,5,17,6 X_4,15,5,16 X_14,22,15,21 X_18,12,19,11 X_20,14,21,13 X_12,20,13,19 X₂₇₃₈ X_6,9,1,10
Gauss code	{1, -10, 2, -5, 4, -11}, {10, -1, 11, -2, 7, -9, 8, -6, 5, -4, 3, -7, 9, -8, 6, -3}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{2}v^{5}-2u^{2}v^{4}+2u^{2}v^{3}-2u^{2}v^{2}+2u^{2}v+uv^{6}-3uv^{5}+5uv^{4}-5uv^{3}+5uv^{2}-3uv+u+2v^{5}-2v^{4}+2v^{3}-2v^{2}+v}{uv^{3}}}$ (db)
Jones polynomial	$-3q^{9/2}+{\frac {2}{q^{9/2}}}+6q^{7/2}-{\frac {5}{q^{7/2}}}-9q^{5/2}+{\frac {7}{q^{5/2}}}+12q^{3/2}-{\frac {11}{q^{3/2}}}+q^{11/2}-{\frac {1}{q^{11/2}}}-13{\sqrt {q}}+{\frac {12}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^{3}z^{5}+z^{5}a^{-3}+4a^{3}z^{3}+3z^{3}a^{-3}+5a^{3}z+2za^{-3}+3a^{3}z^{-1}-az^{7}-z^{7}a^{-1}-5az^{5}-4z^{5}a^{-1}-10az^{3}-4z^{3}a^{-1}-11az+za^{-1}-5az^{-1}+2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-6}-z^{2}a^{-6}+a^{5}z^{7}-5a^{5}z^{5}+3z^{5}a^{-5}+7a^{5}z^{3}-3z^{3}a^{-5}-2a^{5}z+2a^{4}z^{8}-8a^{4}z^{6}+5z^{6}a^{-4}+8a^{4}z^{4}-6z^{4}a^{-4}-a^{4}z^{2}+3z^{2}a^{-4}-a^{-4}+3a^{3}z^{9}-14a^{3}z^{7}+6z^{7}a^{-3}+26a^{3}z^{5}-9z^{5}a^{-3}-28a^{3}z^{3}+7z^{3}a^{-3}+15a^{3}z-2za^{-3}-3a^{3}z^{-1}+a^{2}z^{10}+2a^{2}z^{8}+5z^{8}a^{-2}-18a^{2}z^{6}-6z^{6}a^{-2}+28a^{2}z^{4}-20a^{2}z^{2}+4z^{2}a^{-2}+5a^{2}+6az^{9}+3z^{9}a^{-1}-22az^{7}-z^{7}a^{-1}+36az^{5}-7z^{5}a^{-1}-40az^{3}+5z^{3}a^{-1}+22az+3za^{-1}-5az^{-1}-2a^{-1}z^{-1}+z^{10}+5z^{8}-21z^{6}+27z^{4}-19z^{2}+5$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

12

1

-1

10

2

8

4

1

-3

6

5

2

3

4

7

4

-3

2

6

5

1

0

7

8

1

-2

4

5

-1

-4

3

7

4

-6

2

4

-2

-8

3

-10

1

2

-1

-12

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-6$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{2}$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-1$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=0$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{6}$
$r=1$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=3$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$