L11n192

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_14,9,15,10 X₄₇₅₈ X_16,6,17,5 X_18,16,19,15 X_6,18,1,17 X_19,7,20,22 X_11,20,12,21 X_21,10,22,11 X_2,14,3,13 X_12,4,13,3
Gauss code	{1, -10, 11, -3, 4, -6}, {3, -1, 2, 9, -8, -11, 10, -2, 5, -4, 6, -5, -7, 8, -9, 7}

A Braid Representative

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Polynomial invariants

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

-5

-4

-3

-2

-1

0

1

2

3

4

χ

10

1

8

2

-2

6

4

1

3

4

3

-1

2

5

3

2

0

4

5

1

-2

3

4

-1

-4

2

4

2

-6

1

3

-2

-8

2

-10

1

-1

Integral Khovanov Homology

(db, data source)

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