L11a158

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Link Presentations

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

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{2}v^{4}-4u^{2}v^{3}+6u^{2}v^{2}-4u^{2}v+u^{2}-2uv^{4}+7uv^{3}-9uv^{2}+7uv-2u+v^{4}-4v^{3}+6v^{2}-4v+1}{uv^{2}}}$ (db)
Jones polynomial	$-4q^{9/2}+{\frac {3}{q^{9/2}}}+8q^{7/2}-{\frac {7}{q^{7/2}}}-13q^{5/2}+{\frac {11}{q^{5/2}}}+17q^{3/2}-{\frac {16}{q^{3/2}}}+q^{11/2}-{\frac {1}{q^{11/2}}}-19{\sqrt {q}}+{\frac {18}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+3az^{5}-4z^{5}a^{-1}+z^{5}a^{-3}-3a^{3}z^{3}+9az^{3}-7z^{3}a^{-1}+2z^{3}a^{-3}+a^{5}z-6a^{3}z+8az-5za^{-1}+za^{-3}+a^{5}z^{-1}-2a^{3}z^{-1}+2az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-6}+a^{5}z^{7}-4a^{5}z^{5}+4z^{5}a^{-5}+6a^{5}z^{3}-2z^{3}a^{-5}-4a^{5}z+a^{5}z^{-1}+3a^{4}z^{8}-11a^{4}z^{6}+8z^{6}a^{-4}+13a^{4}z^{4}-7z^{4}a^{-4}-5a^{4}z^{2}+2z^{2}a^{-4}+3a^{3}z^{9}-4a^{3}z^{7}+11z^{7}a^{-3}-13a^{3}z^{5}-14z^{5}a^{-3}+26a^{3}z^{3}+7z^{3}a^{-3}-14a^{3}z-2za^{-3}+2a^{3}z^{-1}+a^{2}z^{10}+10a^{2}z^{8}+10z^{8}a^{-2}-40a^{2}z^{6}-11z^{6}a^{-2}+40a^{2}z^{4}-z^{4}a^{-2}-12a^{2}z^{2}+2z^{2}a^{-2}+a^{2}+8az^{9}+5z^{9}a^{-1}-8az^{7}+8z^{7}a^{-1}-30az^{5}-39z^{5}a^{-1}+44az^{3}+33z^{3}a^{-1}-18az-10za^{-1}+2az^{-1}+a^{-1}z^{-1}+z^{10}+17z^{8}-48z^{6}+34z^{4}-7z^{2}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

12

1

-1

10

3

8

5

1

-4

6

8

3

5

4

9

5

-4

2

10

8

2

0

9

10

1

-2

7

9

-2

-4

5

10

5

-6

2

6

-4

-8

1

5

4

-10

2

-2

-12

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-2$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{7}$
$r=-1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=0$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{10}$
$r=1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=2$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$