L11n114

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_18,8,19,7 X_22,20,5,19 X_20,9,21,10 X_8,21,9,22 X_11,17,12,16 X_17,15,18,14 X_15,11,16,10 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -6, 5, 9, -7, -2, 11, 8, -9, 7, -8, -3, 4, -5, 6, -4}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{4}-2t(2)^{3}+t(2)^{2}-2t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$7q^{9/2}-9q^{7/2}+9q^{5/2}-{\frac {1}{q^{5/2}}}-9q^{3/2}+{\frac {2}{q^{3/2}}}+2q^{13/2}-5q^{11/2}+7{\sqrt {q}}-{\frac {5}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$za^{-7}+a^{-7}z^{-1}-z^{5}a^{-5}-4z^{3}a^{-5}-6za^{-5}-4a^{-5}z^{-1}+z^{7}a^{-3}+5z^{5}a^{-3}+10z^{3}a^{-3}+12za^{-3}+6a^{-3}z^{-1}-2z^{5}a^{-1}+az^{3}-8z^{3}a^{-1}+3az-10za^{-1}+2az^{-1}-5a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{9}a^{-1}-z^{9}a^{-3}-7z^{8}a^{-2}-5z^{8}a^{-4}-2z^{8}-az^{7}-3z^{7}a^{-1}-10z^{7}a^{-3}-8z^{7}a^{-5}+23z^{6}a^{-2}+10z^{6}a^{-4}-5z^{6}a^{-6}+8z^{6}+5az^{5}+26z^{5}a^{-1}+47z^{5}a^{-3}+25z^{5}a^{-5}-z^{5}a^{-7}-16z^{4}a^{-2}+2z^{4}a^{-4}+9z^{4}a^{-6}-9z^{4}-9az^{3}-41z^{3}a^{-1}-58z^{3}a^{-3}-31z^{3}a^{-5}-5z^{3}a^{-7}-8z^{2}a^{-4}-9z^{2}a^{-6}-3z^{2}a^{-8}+2z^{2}+7az+24za^{-1}+31za^{-3}+18za^{-5}+4za^{-7}+a^{-2}+3a^{-4}+3a^{-6}+a^{-8}+1-2az^{-1}-5a^{-1}z^{-1}-6a^{-3}z^{-1}-4a^{-5}z^{-1}-a^{-7}z^{-1}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

χ

14

2

-2

12

3

10

4

2

-2

8

5

3

2

6

4

0

4

5

0

2

4

6

2

0

1

3

-2

1

4

3

-4

1

-1

-6

1

Integral Khovanov Homology

(db, data source)

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