L11n277

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₃₈₄₉ X_2,14,3,13 X_14,7,15,8 X_9,18,10,19 X_22,17,11,18 X_20,11,21,12 X_16,21,17,22 X_15,1,16,4 X_19,10,20,5
Gauss code	{1, -4, -3, 10}, {-2, -1, 5, 3, -6, 11}, {8, 2, 4, -5, -10, -9, 7, 6, -11, -8, 9, -7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {uvw^{4}-uvw^{3}+uvw^{2}-uvw+uw-v^{2}w^{3}+vw^{3}-vw^{2}+vw-v}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$2q^{-2}-q^{-3}+4q^{-4}-3q^{-5}+4q^{-6}-3q^{-7}+2q^{-8}-q^{-9}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{10}z^{-2}-a^{10}+z^{4}a^{8}+5z^{2}a^{8}+4a^{8}z^{-2}+8a^{8}-z^{6}a^{6}-6z^{4}a^{6}-14z^{2}a^{6}-5a^{6}z^{-2}-15a^{6}+2z^{4}a^{4}+8z^{2}a^{4}+2a^{4}z^{-2}+8a^{4}$ (db)
Kauffman polynomial	$z^{3}a^{11}-2za^{11}+a^{11}z^{-1}+2z^{4}a^{10}-3z^{2}a^{10}-a^{10}z^{-2}+2a^{10}+z^{7}a^{9}-5z^{5}a^{9}+14z^{3}a^{9}-13za^{9}+5a^{9}z^{-1}+z^{8}a^{8}-6z^{6}a^{8}+18z^{4}a^{8}-21z^{2}a^{8}-4a^{8}z^{-2}+13a^{8}+2z^{7}a^{7}-10z^{5}a^{7}+26z^{3}a^{7}-27za^{7}+9a^{7}z^{-1}+z^{8}a^{6}-6z^{6}a^{6}+19z^{4}a^{6}-29z^{2}a^{6}-5a^{6}z^{-2}+20a^{6}+z^{7}a^{5}-5z^{5}a^{5}+13z^{3}a^{5}-16za^{5}+5a^{5}z^{-1}+3z^{4}a^{4}-11z^{2}a^{4}-2a^{4}z^{-2}+10a^{4}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

χ

-3

2

-5

1

2

1

-7

3

-9

1

2

1

-11

3

2

1

-13

1

-15

2

3

-1

-17

1

-19

1

-1

Integral Khovanov Homology

(db, data source)

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