L11n367

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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_13,20,14,21 X_19,9,20,8 X_9,19,10,18 X_15,22,16,17 X_17,16,18,5 X_21,14,22,15 X₂₅₃₆ X_4,12,1,11
Gauss code	{1, -10, 2, -11}, {-8, 6, -5, 4, -9, 7}, {10, -1, 3, 5, -6, -2, 11, -3, -4, 9, -7, 8}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(w-1)\left(v^{2}w-1\right)}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$-q^{4}+2q^{3}+2q^{-3}-2q^{2}-q^{-2}+3q+3q^{-1}-2$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{4}z^{-2}+a^{4}-a^{2}z^{4}-z^{4}a^{-2}-4a^{2}z^{2}-3z^{2}a^{-2}-2a^{2}z^{-2}-4a^{2}-a^{-2}+z^{6}+5z^{4}+7z^{2}+z^{-2}+4$ (db)
Kauffman polynomial	$-a^{4}z^{2}-a^{4}z^{-2}+3a^{4}+z^{7}a^{-3}-5z^{5}a^{-3}+6z^{3}a^{-3}-2a^{3}z-za^{-3}+2a^{3}z^{-1}+a^{2}z^{8}+2z^{8}a^{-2}-6a^{2}z^{6}-11z^{6}a^{-2}+12a^{2}z^{4}+17z^{4}a^{-2}-13a^{2}z^{2}-9z^{2}a^{-2}-2a^{2}z^{-2}+7a^{2}+2a^{-2}+az^{9}+z^{9}a^{-1}-5az^{7}-4z^{7}a^{-1}+6az^{5}+z^{5}a^{-1}-az^{3}+5z^{3}a^{-1}-4az-3za^{-1}+2az^{-1}+3z^{8}-17z^{6}+29z^{4}-21z^{2}-z^{-2}+7$ (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

χ

9

1

-1

7

1

5

1

0

3

2

1

-1

3

2

1

-3

1

2

-5

2

1

-7

2

Integral Khovanov Homology

(db, data source)

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