L11n115

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_20,8,21,7 X_22,17,5,18 X_18,21,19,22 X_13,17,14,16 X_9,15,10,14 X_15,11,16,10 X_8,20,9,19 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, -7, 8, 11, -2, -6, 7, -8, 6, 4, -5, 9, -3, 5, -4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(2)^{5}+t(1)t(2)^{4}-4t(2)^{4}-2t(1)t(2)^{3}+3t(2)^{3}+3t(1)t(2)^{2}-2t(2)^{2}-4t(1)t(2)+t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{7/2}-3q^{5/2}+5q^{3/2}-7{\sqrt {q}}+{\frac {7}{\sqrt {q}}}-{\frac {7}{q^{3/2}}}+{\frac {6}{q^{5/2}}}-{\frac {5}{q^{7/2}}}+{\frac {2}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$za^{5}+2a^{5}z^{-1}-3z^{3}a^{3}-8za^{3}-5a^{3}z^{-1}+2z^{5}a+8z^{3}a+11za+6az^{-1}-3z^{3}a^{-1}-7za^{-1}-4a^{-1}z^{-1}+za^{-3}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$a^{5}z^{7}-5a^{5}z^{5}+9a^{5}z^{3}-7a^{5}z+2a^{5}z^{-1}+2a^{4}z^{8}-8a^{4}z^{6}+8a^{4}z^{4}-a^{4}z^{2}+z^{2}a^{-4}-a^{4}-a^{-4}+a^{3}z^{9}+2a^{3}z^{7}-23a^{3}z^{5}+36a^{3}z^{3}+3z^{3}a^{-3}-22a^{3}z-3za^{-3}+5a^{3}z^{-1}+a^{-3}z^{-1}+6a^{2}z^{8}-22a^{2}z^{6}+2z^{6}a^{-2}+18a^{2}z^{4}-5z^{4}a^{-2}-2a^{2}z^{2}+8z^{2}a^{-2}-a^{2}-3a^{-2}+az^{9}+6az^{7}+5z^{7}a^{-1}-38az^{5}-20z^{5}a^{-1}+54az^{3}+30z^{3}a^{-1}-30az-18za^{-1}+6az^{-1}+4a^{-1}z^{-1}+4z^{8}-12z^{6}+5z^{4}+6z^{2}-3$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

8

1

-1

6

2

4

3

1

-2

2

4

2

0

4

0

-2

3

4

1

0

-4

3

4

1

-6

2

3

-1

-8

1

4

3

-10

1

-1

-12

1

Integral Khovanov Homology

(db, data source)

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