L11n111

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_3,12,4,13 X_7,18,8,19 X_19,22,20,5 X_9,21,10,20 X_21,9,22,8 X_16,11,17,12 X_14,17,15,18 X_10,15,11,16 X₂₅₃₆ X_13,4,14,1
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 $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(2)^{5}-2t(1)t(2)^{4}-t(2)^{4}-t(1)t(2)^{3}+2t(2)^{3}+2t(1)t(2)^{2}-t(2)^{2}-t(1)t(2)-2t(2)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}-{\frac {2}{q^{5/2}}}+{\frac {1}{q^{7/2}}}-{\frac {1}{q^{11/2}}}+{\frac {1}{q^{13/2}}}-{\frac {2}{q^{15/2}}}+{\frac {2}{q^{17/2}}}-{\frac {2}{q^{19/2}}}+{\frac {1}{q^{21/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{11}z^{-1}+4za^{9}+4a^{9}z^{-1}-z^{5}a^{7}-7z^{3}a^{7}-11za^{7}-6a^{7}z^{-1}+z^{7}a^{5}+6z^{5}a^{5}+11z^{3}a^{5}+11za^{5}+5a^{5}z^{-1}-z^{5}a^{3}-5z^{3}a^{3}-6za^{3}-2a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{12}z^{6}-4a^{12}z^{4}+3a^{12}z^{2}-a^{12}+2a^{11}z^{7}-9a^{11}z^{5}+9a^{11}z^{3}-4a^{11}z+a^{11}z^{-1}+a^{10}z^{8}-3a^{10}z^{6}-4a^{10}z^{4}+9a^{10}z^{2}-3a^{10}+3a^{9}z^{7}-18a^{9}z^{5}+29a^{9}z^{3}-18a^{9}z+4a^{9}z^{-1}+a^{8}z^{8}-5a^{8}z^{6}+a^{8}z^{4}+7a^{8}z^{2}-3a^{8}+3a^{7}z^{7}-22a^{7}z^{5}+42a^{7}z^{3}-28a^{7}z+6a^{7}z^{-1}+a^{6}z^{8}-6a^{6}z^{6}+6a^{6}z^{4}+a^{6}z^{2}-a^{6}+3a^{5}z^{7}-19a^{5}z^{5}+33a^{5}z^{3}-22a^{5}z+5a^{5}z^{-1}+a^{4}z^{8}-5a^{4}z^{6}+5a^{4}z^{4}-a^{4}+a^{3}z^{7}-6a^{3}z^{5}+11a^{3}z^{3}-8a^{3}z+2a^{3}z^{-1}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

0

1

-2

0

-4

2

1

-6

1

-8

1

-1

-10

2

1

-12

2

3

1

0

-14

1

-16

1

2

1

0

-18

1

0

-20

1

-22

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-6$	$i=-4$	$i=-2$
$r=-9$		${\mathbb {Z} }$
$r=-8$		${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-7$		${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$		${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{2}$
$r=-3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=0$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{2}$
$r=1$	${\mathbb {Z} }$
$r=2$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$