L11n433

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_14,4,15,3 X_12,14,7,13 X₂₇₃₈ X_22,10,13,9 X_6,22,1,21 X_20,16,21,15 X_5,17,6,16 X_11,19,12,18 X_17,11,18,10 X_19,5,20,4
Gauss code	{1, -4, 2, 11, -8, -6}, {4, -1, 5, 10, -9, -3}, {3, -2, 7, 8, -10, 9, -11, -7, 6, -5}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(uvw+1)\left(uvw^{2}-uw^{2}+u-vw^{2}+v-1\right)}{uvw^{3/2}}}$ (db)
Jones polynomial	$q^{10}-q^{9}+q^{8}-q^{7}+q^{6}+q^{5}+2q^{3}-q^{2}+q$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$-z^{8}a^{-6}+z^{6}a^{-4}-7z^{6}a^{-6}+z^{6}a^{-8}+6z^{4}a^{-4}-16z^{4}a^{-6}+6z^{4}a^{-8}+10z^{2}a^{-4}-17z^{2}a^{-6}+8z^{2}a^{-8}-z^{2}a^{-10}+6a^{-4}-9a^{-6}+3a^{-8}+a^{-4}z^{-2}-2a^{-6}z^{-2}+a^{-8}z^{-2}$ (db)
Kauffman polynomial	$z^{4}a^{-12}-3z^{2}a^{-12}+z^{5}a^{-11}-3z^{3}a^{-11}+z^{6}a^{-10}-5z^{4}a^{-10}+5z^{2}a^{-10}-2a^{-10}+z^{7}a^{-9}-6z^{5}a^{-9}+7z^{3}a^{-9}+2z^{8}a^{-8}-13z^{6}a^{-8}+22z^{4}a^{-8}-11z^{2}a^{-8}-a^{-8}z^{-2}+3a^{-8}+z^{9}a^{-7}-5z^{7}a^{-7}+z^{5}a^{-7}+13z^{3}a^{-7}-9za^{-7}+2a^{-7}z^{-1}+3z^{8}a^{-6}-21z^{6}a^{-6}+44z^{4}a^{-6}-35z^{2}a^{-6}-2a^{-6}z^{-2}+11a^{-6}+z^{9}a^{-5}-6z^{7}a^{-5}+8z^{5}a^{-5}+3z^{3}a^{-5}-9za^{-5}+2a^{-5}z^{-1}+z^{8}a^{-4}-7z^{6}a^{-4}+16z^{4}a^{-4}-16z^{2}a^{-4}-a^{-4}z^{-2}+7a^{-4}$ (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

7

8

χ

21

1

19

1

0

17

1

0

15

1

2

1

0

13

1

2

1

0

11

1

2

9

1

3

1

7

1

2

5

1

2

1

3

0

1

Integral Khovanov Homology

(db, data source)

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