L11n45

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_17,1,18,4 X_5,14,6,15 X₃₈₄₉ X_9,18,10,19 X_11,20,12,21 X_13,22,14,5 X_19,10,20,11 X_21,12,22,13 X_2,16,3,15
Gauss code	{1, -11, -5, 3}, {-4, -1, 2, 5, -6, 9, -7, 10, -8, 4, 11, -2, -3, 6, -9, 7, -10, 8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}-t(2)+1\right)\left(t(2)^{2}+t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {2}{q^{5/2}}}+{\frac {1}{q^{7/2}}}-{\frac {3}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {4}{q^{13/2}}}+{\frac {4}{q^{15/2}}}-{\frac {3}{q^{17/2}}}+{\frac {2}{q^{19/2}}}-{\frac {1}{q^{21/2}}}+{\frac {1}{q^{23/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-z^{5}a^{9}-5z^{3}a^{9}-7za^{9}-3a^{9}z^{-1}+z^{7}a^{7}+7z^{5}a^{7}+18z^{3}a^{7}+20za^{7}+7a^{7}z^{-1}-2z^{5}a^{5}-10z^{3}a^{5}-13za^{5}-4a^{5}z^{-1}$ (db)
Kauffman polynomial	$a^{14}z^{4}-3a^{14}z^{2}+a^{14}+a^{13}z^{5}-2a^{13}z^{3}+a^{12}z^{6}-2a^{12}z^{4}+a^{12}z^{2}+a^{11}z^{7}-3a^{11}z^{5}+4a^{11}z^{3}+a^{10}z^{8}-4a^{10}z^{6}+6a^{10}z^{4}+a^{9}z^{9}-6a^{9}z^{7}+15a^{9}z^{5}-16a^{9}z^{3}+10a^{9}z-3a^{9}z^{-1}+2a^{8}z^{8}-11a^{8}z^{6}+24a^{8}z^{4}-23a^{8}z^{2}+7a^{8}+a^{7}z^{9}-7a^{7}z^{7}+22a^{7}z^{5}-36a^{7}z^{3}+26a^{7}z-7a^{7}z^{-1}+a^{6}z^{8}-6a^{6}z^{6}+15a^{6}z^{4}-19a^{6}z^{2}+7a^{6}+3a^{5}z^{5}-14a^{5}z^{3}+16a^{5}z-4a^{5}z^{-1}$ (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

χ

-4

2

-6

1

2

1

-8

2

-10

1

0

-12

3

2

1

-14

1

0

-16

2

3

-1

-18

1

-20

1

2

-1

-22

0

-24

1

-1

Integral Khovanov Homology

(db, data source)

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