L11n145

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_18,11,19,12 X_10,4,11,3 X_2,17,3,18 X_12,5,13,6 X₆₇₁₈ X_16,10,17,9 X_15,20,16,21 X_13,22,14,7 X_21,14,22,15 X_4,20,5,19
Gauss code	{1, -4, 3, -11, 5, -6}, {6, -1, 7, -3, 2, -5, -9, 10, -8, -7, 4, -2, 11, 8, -10, 9}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)^{2}t(2)^{4}-t(1)t(2)^{4}-3t(1)^{2}t(2)^{3}+4t(1)t(2)^{3}-t(2)^{3}+3t(1)^{2}t(2)^{2}-5t(1)t(2)^{2}+3t(2)^{2}-t(1)^{2}t(2)+4t(1)t(2)-3t(2)-t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{3/2}-4{\sqrt {q}}+{\frac {6}{\sqrt {q}}}-{\frac {9}{q^{3/2}}}+{\frac {10}{q^{5/2}}}-{\frac {11}{q^{7/2}}}+{\frac {9}{q^{9/2}}}-{\frac {7}{q^{11/2}}}+{\frac {4}{q^{13/2}}}-{\frac {1}{q^{15/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{3}z^{7}+a^{5}z^{5}-4a^{3}z^{5}+az^{5}+2a^{5}z^{3}-4a^{3}z^{3}+2az^{3}-az+a^{3}z^{-1}-az^{-1}$ (db)
Kauffman polynomial	$-z^{3}a^{9}-4z^{4}a^{8}+2z^{2}a^{8}-z^{7}a^{7}-3z^{5}a^{7}+2z^{3}a^{7}-3z^{8}a^{6}+7z^{6}a^{6}-12z^{4}a^{6}+6z^{2}a^{6}-2z^{9}a^{5}+2z^{7}a^{5}-8z^{8}a^{4}+23z^{6}a^{4}-20z^{4}a^{4}+6z^{2}a^{4}-2z^{9}a^{3}-z^{7}a^{3}+15z^{5}a^{3}-10z^{3}a^{3}-za^{3}+a^{3}z^{-1}-5z^{8}a^{2}+15z^{6}a^{2}-10z^{4}a^{2}+2z^{2}a^{2}-a^{2}-4z^{7}a+12z^{5}a-7z^{3}a-za+az^{-1}-z^{6}+2z^{4}$ (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

χ

4

1

-1

2

3

0

3

1

-2

6

3

-4

5

4

-1

-6

6

5

1

-8

4

6

2

-10

3

5

-2

-12

1

4

3

-14

3

-3

-16

1

Integral Khovanov Homology

(db, data source)

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