L11n400

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_15,22,16,19 X_7,20,8,21 X_19,8,20,9 X_18,14,5,13 X_14,12,15,11 X_12,18,13,17 X_21,16,22,17 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {-5, 4, -9, 3}, {10, -1, -4, 5, 11, -2, 7, -8, 6, -7, -3, 9, 8, -6}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(3)-1)\left(t(3)^{4}+t(1)t(2)t(3)^{3}-t(2)t(3)^{3}+t(3)^{3}+t(1)t(3)^{2}-t(1)t(2)t(3)^{2}+t(2)t(3)^{2}-t(3)^{2}-t(1)t(3)+t(1)t(2)t(3)+t(3)+t(1)t(2)\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{5/2}}}$ (db)
Jones polynomial	$-q^{2}+3q-5+6q^{-1}-6q^{-2}+6q^{-3}-4q^{-4}+3q^{-5}+q^{-6}+q^{-8}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$2a^{8}z^{-2}+a^{8}-2z^{2}a^{6}-5a^{6}z^{-2}-7a^{6}+3z^{2}a^{4}+4a^{4}z^{-2}+6a^{4}+z^{6}a^{2}+4z^{4}a^{2}+5z^{2}a^{2}-a^{2}z^{-2}+a^{2}-z^{4}-2z^{2}-1$ (db)
Kauffman polynomial	$z^{8}a^{8}-8z^{6}a^{8}+21z^{4}a^{8}-24z^{2}a^{8}-2a^{8}z^{-2}+12a^{8}+z^{7}a^{7}-10z^{5}a^{7}+25z^{3}a^{7}-21za^{7}+5a^{7}z^{-1}+z^{8}a^{6}-11z^{6}a^{6}+35z^{4}a^{6}-41z^{2}a^{6}-5a^{6}z^{-2}+23a^{6}+2z^{7}a^{5}-15z^{5}a^{5}+42z^{3}a^{5}-35za^{5}+9a^{5}z^{-1}+z^{8}a^{4}-3z^{6}a^{4}+9z^{4}a^{4}-12z^{2}a^{4}-4a^{4}z^{-2}+12a^{4}+4z^{7}a^{3}-11z^{5}a^{3}+17z^{3}a^{3}-15za^{3}+5a^{3}z^{-1}+z^{8}a^{2}+3z^{6}a^{2}-12z^{4}a^{2}+8z^{2}a^{2}-a^{2}z^{-2}-a^{2}+3z^{7}a-5z^{5}a-2z^{3}a+az^{-1}+3z^{6}-7z^{4}+3z^{2}-1+z^{5}a^{-1}-2z^{3}a^{-1}+za^{-1}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

5

1

-1

3

2

1

3

1

-2

-1

3

2

1

-3

4

0

-5

1

3

2

0

-7

2

4

2

-9

1

3

-1

-11

4

-13

1

-15

1

-17

1

Integral Khovanov Homology

(db, data source)

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