L10n110

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(3)^{2}-t(1)t(2)t(3)^{2}-t(1)t(4)t(3)^{2}+t(1)t(2)t(4)t(3)^{2}+t(4)t(3)^{2}-t(1)t(4)^{2}t(3)-t(2)t(4)^{2}t(3)+t(4)^{2}t(3)-t(1)t(3)+t(1)t(2)t(3)-t(2)t(3)+2t(1)t(4)t(3)-2t(1)t(2)t(4)t(3)+2t(2)t(4)t(3)-2t(4)t(3)+t(2)t(4)^{2}-t(4)^{2}+t(1)t(2)t(4)-t(2)t(4)+t(4)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)t(4)}}$ (db)
Jones polynomial	$-6q^{9/2}+6q^{7/2}-10q^{5/2}+7q^{3/2}-{\frac {3}{q^{3/2}}}-q^{13/2}+3q^{11/2}-8{\sqrt {q}}+{\frac {4}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{-5}-a^{-5}z^{-3}-za^{-5}-2a^{-5}z^{-1}+z^{5}a^{-3}+3z^{3}a^{-3}+3a^{-3}z^{-3}+7za^{-3}+7a^{-3}z^{-1}-4z^{3}a^{-1}+az^{-3}-3a^{-1}z^{-3}+3az-9za^{-1}+3az^{-1}-8a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2z^{8}a^{-2}-2z^{8}a^{-4}-5z^{7}a^{-1}-9z^{7}a^{-3}-4z^{7}a^{-5}-z^{6}a^{-2}-z^{6}a^{-4}-3z^{6}a^{-6}-3z^{6}+17z^{5}a^{-1}+27z^{5}a^{-3}+9z^{5}a^{-5}-z^{5}a^{-7}+11z^{4}a^{-2}+11z^{4}a^{-4}+6z^{4}a^{-6}+6z^{4}-6az^{3}-33z^{3}a^{-1}-36z^{3}a^{-3}-7z^{3}a^{-5}+2z^{3}a^{-7}-23z^{2}a^{-2}-13z^{2}a^{-4}-z^{2}a^{-6}-11z^{2}+11az+27za^{-1}+25za^{-3}+8za^{-5}-za^{-7}+19a^{-2}+10a^{-4}+10-5az^{-1}-12a^{-1}z^{-1}-12a^{-3}z^{-1}-5a^{-5}z^{-1}-6a^{-2}z^{-2}-3a^{-4}z^{-2}-3z^{-2}+az^{-3}+3a^{-1}z^{-3}+3a^{-3}z^{-3}+a^{-5}z^{-3}$ (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		\

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

2

-2

10

4

1

3

8

4

0

6

2

4

3

6

3

2

5

4

1

0

1

5

4

-2

2

3

-1

-4

3

Integral Khovanov Homology

(db, data source)

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