L10a148

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,4,15,3 X_16,5,17,6 X_12,15,5,16 X_8,20,9,19 X_18,8,19,7 X_20,10,13,9 X_10,14,11,13 X_2,11,3,12 X_4,18,1,17
Gauss code	{1, -9, 2, -10}, {3, -1, 6, -5, 7, -8, 9, -4}, {8, -2, 4, -3, 10, -6, 5, -7}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {uv^{3}w^{3}-uv^{3}w^{2}-uv^{2}w^{3}+3uv^{2}w^{2}-uv^{2}w-2uvw^{2}+3uvw-uv-uw+u-v^{3}w^{3}+v^{3}w^{2}+v^{2}w^{3}-3v^{2}w^{2}+2v^{2}w+vw^{2}-3vw+v+w-1}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$q^{7}-3q^{6}+5q^{5}-7q^{4}+10q^{3}+q^{-3}-9q^{2}-3q^{-2}+10q+5q^{-1}-6$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-4}+4z^{4}a^{-4}+4z^{2}a^{-4}+a^{-4}z^{-2}+a^{-4}-z^{8}a^{-2}-6z^{6}a^{-2}-12z^{4}a^{-2}-9z^{2}a^{-2}-2a^{-2}z^{-2}-3a^{-2}+z^{6}+4z^{4}+4z^{2}+z^{-2}+2$ (db)
Kauffman polynomial	$2z^{9}a^{-1}+2z^{9}a^{-3}+8z^{8}a^{-2}+4z^{8}a^{-4}+4z^{8}+3az^{7}-3z^{7}a^{-1}-2z^{7}a^{-3}+4z^{7}a^{-5}+a^{2}z^{6}-30z^{6}a^{-2}-10z^{6}a^{-4}+4z^{6}a^{-6}-15z^{6}-10az^{5}-5z^{5}a^{-1}-3z^{5}a^{-3}-5z^{5}a^{-5}+3z^{5}a^{-7}-3a^{2}z^{4}+41z^{4}a^{-2}+15z^{4}a^{-4}-5z^{4}a^{-6}+z^{4}a^{-8}+17z^{4}+6az^{3}+7z^{3}a^{-1}+7z^{3}a^{-3}+2z^{3}a^{-5}-4z^{3}a^{-7}+a^{2}z^{2}-22z^{2}a^{-2}-10z^{2}a^{-4}+z^{2}a^{-6}-z^{2}a^{-8}-9z^{2}-3za^{-1}-3za^{-3}+5a^{-2}+3a^{-4}+3+2a^{-1}z^{-1}+2a^{-3}z^{-1}-2a^{-2}z^{-2}-a^{-4}z^{-2}-z^{-2}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

15

1

13

2

-2

11

3

1

2

9

5

3

-2

7

5

2

3

5

4

5

1

3

6

5

1

3

7

4

-1

2

3

-1

-3

1

3

2

-5

2

-2

-7

1

Integral Khovanov Homology

(db, data source)

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