L11n366

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,4,11,3 X_12,8,13,7 X_15,17,16,22 X_18,10,19,9 X_8,18,9,17 X_13,21,14,20 X_21,15,22,14 X_19,5,20,16 X₂₅₃₆ X_4,12,1,11
Gauss code	{1, -10, 2, -11}, {6, -5, -9, 7, -8, 4}, {10, -1, 3, -6, 5, -2, 11, -3, -7, 8, -4, 9}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(w-1)\left(v^{2}w^{3}-1\right)}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$q^{9}-q^{8}+2q^{7}-2q^{6}+3q^{5}-2q^{4}+3q^{3}-q^{2}+q$ (db)
Signature	6 (db)
HOMFLY-PT polynomial	$-z^{8}a^{-6}+z^{6}a^{-4}-7z^{6}a^{-6}+z^{6}a^{-8}+6z^{4}a^{-4}-17z^{4}a^{-6}+6z^{4}a^{-8}+11z^{2}a^{-4}-20z^{2}a^{-6}+10z^{2}a^{-8}-z^{2}a^{-10}+7a^{-4}-12a^{-6}+6a^{-8}-a^{-10}+a^{-4}z^{-2}-2a^{-6}z^{-2}+a^{-8}z^{-2}$ (db)
Kauffman polynomial	$a^{-12}-za^{-11}+z^{4}a^{-10}-5z^{2}a^{-10}+2a^{-10}+2z^{7}a^{-9}-10z^{5}a^{-9}+11z^{3}a^{-9}-3za^{-9}+3z^{8}a^{-8}-18z^{6}a^{-8}+33z^{4}a^{-8}-26z^{2}a^{-8}-a^{-8}z^{-2}+9a^{-8}+z^{9}a^{-7}-3z^{7}a^{-7}-6z^{5}a^{-7}+19z^{3}a^{-7}-12za^{-7}+2a^{-7}z^{-1}+4z^{8}a^{-6}-25z^{6}a^{-6}+49z^{4}a^{-6}-39z^{2}a^{-6}-2a^{-6}z^{-2}+15a^{-6}+z^{9}a^{-5}-5z^{7}a^{-5}+4z^{5}a^{-5}+8z^{3}a^{-5}-10za^{-5}+2a^{-5}z^{-1}+z^{8}a^{-4}-7z^{6}a^{-4}+17z^{4}a^{-4}-18z^{2}a^{-4}-a^{-4}z^{-2}+8a^{-4}$ (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		\

-2

-1

0

1

2

3

4

5

6

χ

19

1

17

2

0

15

1

13

2

0

11

1

9

1

2

1

7

2

1

5

1

3

2

3

0

1

Integral Khovanov Homology

(db, data source)

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