L10n30

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_3,10,4,11 X_15,5,16,20 X_7,17,8,16 X_17,12,18,13 X_9,14,10,15 X_13,18,14,19 X_19,9,20,8 X₂₅₃₆ X_11,4,12,1
Gauss code	{1, -9, -2, 10}, {9, -1, -4, 8, -6, 2, -10, 5, -7, 6, -3, 4, -5, 7, -8, 3}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)t(2)^{5}-3t(1)t(2)^{4}+t(2)^{4}+4t(1)t(2)^{3}-2t(2)^{3}-2t(1)t(2)^{2}+4t(2)^{2}+t(1)t(2)-3t(2)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	${\frac {6}{q^{9/2}}}-{\frac {8}{q^{7/2}}}+{\frac {7}{q^{5/2}}}+q^{3/2}-{\frac {7}{q^{3/2}}}+{\frac {2}{q^{13/2}}}-{\frac {5}{q^{11/2}}}-3{\sqrt {q}}+{\frac {5}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-za^{7}-a^{7}z^{-1}+z^{5}a^{5}+4z^{3}a^{5}+6za^{5}+4a^{5}z^{-1}-z^{7}a^{3}-5z^{5}a^{3}-9z^{3}a^{3}-9za^{3}-4a^{3}z^{-1}+z^{5}a+3z^{3}a+2za+az^{-1}$ (db)
Kauffman polynomial	$-3z^{2}a^{8}+a^{8}-z^{5}a^{7}-5z^{3}a^{7}+4za^{7}-a^{7}z^{-1}-5z^{6}a^{6}+11z^{4}a^{6}-13z^{2}a^{6}+4a^{6}-6z^{7}a^{5}+18z^{5}a^{5}-24z^{3}a^{5}+16za^{5}-4a^{5}z^{-1}-2z^{8}a^{4}-3z^{6}a^{4}+21z^{4}a^{4}-21z^{2}a^{4}+7a^{4}-9z^{7}a^{3}+29z^{5}a^{3}-28z^{3}a^{3}+15za^{3}-4a^{3}z^{-1}-2z^{8}a^{2}+z^{6}a^{2}+13z^{4}a^{2}-14z^{2}a^{2}+4a^{2}-3z^{7}a+10z^{5}a-9z^{3}a+3za-az^{-1}-z^{6}+3z^{4}-3z^{2}+1$ (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

χ

4

1

-1

2

0

3

1

-2

4

2

-4

4

0

-6

4

3

1

-8

2

4

2

-10

3

4

-1

-12

3

-14

2

-2

Integral Khovanov Homology

(db, data source)

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