L11n64

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_14,8,15,7 X_15,20,16,21 X_11,18,12,19 X_19,12,20,13 X_17,22,18,5 X_21,16,22,17 X_8,14,9,13 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, -5, 6, 9, -3, -4, 8, -7, 5, -6, 4, -8, 7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{5}-2uv^{4}+uv^{2}-u-v^{5}+v^{3}-2v+1}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{\sqrt {q}}}+{\frac {2}{q^{3/2}}}-{\frac {3}{q^{5/2}}}+{\frac {3}{q^{7/2}}}-{\frac {3}{q^{9/2}}}+{\frac {1}{q^{11/2}}}-{\frac {2}{q^{13/2}}}+{\frac {1}{q^{17/2}}}-{\frac {1}{q^{19/2}}}+{\frac {1}{q^{21/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-a^{11}z^{-1}+3a^{9}z+2a^{9}z^{-1}-a^{7}z^{5}-5a^{7}z^{3}-4a^{7}z+a^{5}z^{7}+5a^{5}z^{5}+6a^{5}z^{3}+2a^{5}z-a^{5}z^{-1}-a^{3}z^{5}-4a^{3}z^{3}-3a^{3}z$ (db)
Kauffman polynomial	$-z^{6}a^{12}+5z^{4}a^{12}-6z^{2}a^{12}+2a^{12}-z^{7}a^{11}+5z^{5}a^{11}-5z^{3}a^{11}+2za^{11}-a^{11}z^{-1}-z^{6}a^{10}+8z^{4}a^{10}-12z^{2}a^{10}+5a^{10}+2z^{5}a^{9}-3z^{3}a^{9}+3za^{9}-2a^{9}z^{-1}-z^{8}a^{8}+6z^{6}a^{8}-6z^{4}a^{8}-z^{2}a^{8}+3a^{8}-z^{9}a^{7}+5z^{7}a^{7}-5z^{5}a^{7}+3z^{3}a^{7}-3za^{7}-3z^{8}a^{6}+16z^{6}a^{6}-22z^{4}a^{6}+10z^{2}a^{6}-a^{6}-z^{9}a^{5}+3z^{7}a^{5}+3z^{5}a^{5}-6z^{3}a^{5}-za^{5}+a^{5}z^{-1}-2z^{8}a^{4}+10z^{6}a^{4}-13z^{4}a^{4}+5z^{2}a^{4}-z^{7}a^{3}+5z^{5}a^{3}-7z^{3}a^{3}+3za^{3}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

0

1

-2

1

-1

-4

2

1

-6

2

0

-8

1

2

1

0

-10

1

2

-12

1

4

2

1

-14

1

2

-16

1

2

-1

-18

1

0

-20

0

-22

1

-1

Integral Khovanov Homology

(db, data source)

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