L11n150

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_18,9,19,10 X₆₇₁₈ X_22,19,7,20 X_12,5,13,6 X_3,10,4,11 X_15,5,16,4 X_11,16,12,17 X_20,13,21,14 X_14,21,15,22 X_2,18,3,17
Gauss code	{1, -11, -6, 7, 5, -3}, {3, -1, 2, 6, -8, -5, 9, -10, -7, 8, 11, -2, 4, -9, 10, -4}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{2}v^{2}-u^{2}v-3uv^{2}+5uv-3u-v+1}{uv}}$ (db)
Jones polynomial	$-{\frac {1}{q^{3/2}}}+{\frac {1}{q^{5/2}}}-{\frac {4}{q^{7/2}}}+{\frac {4}{q^{9/2}}}-{\frac {5}{q^{11/2}}}+{\frac {5}{q^{13/2}}}-{\frac {4}{q^{15/2}}}+{\frac {3}{q^{17/2}}}-{\frac {2}{q^{19/2}}}+{\frac {1}{q^{21/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-z^{3}a^{9}-2za^{9}-a^{9}z^{-1}+z^{5}a^{7}+4z^{3}a^{7}+6za^{7}+2a^{7}z^{-1}-2z^{3}a^{5}-3za^{5}-z^{3}a^{3}-2za^{3}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$-z^{6}a^{12}+4z^{4}a^{12}-3z^{2}a^{12}-2z^{7}a^{11}+8z^{5}a^{11}-7z^{3}a^{11}+za^{11}-2z^{8}a^{10}+8z^{6}a^{10}-9z^{4}a^{10}+6z^{2}a^{10}-2a^{10}-z^{9}a^{9}+3z^{7}a^{9}-2z^{5}a^{9}+2z^{3}a^{9}-2za^{9}+a^{9}z^{-1}-3z^{8}a^{8}+14z^{6}a^{8}-25z^{4}a^{8}+20z^{2}a^{8}-5a^{8}-z^{9}a^{7}+5z^{7}a^{7}-12z^{5}a^{7}+13z^{3}a^{7}-8za^{7}+2a^{7}z^{-1}-z^{8}a^{6}+5z^{6}a^{6}-13z^{4}a^{6}+11z^{2}a^{6}-3a^{6}-2z^{5}a^{5}+3z^{3}a^{5}-3za^{5}-z^{4}a^{4}+a^{4}-z^{3}a^{3}+2za^{3}-a^{3}z^{-1}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-2

1

-4

1

0

-6

3

-8

2

0

-10

3

2

1

-12

2

0

-14

2

3

-1

-16

1

2

1

-18

1

2

-1

-20

1

-22

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$
$r=-9$	${\mathbb {Z} }$
$r=-8$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-5$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-4$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-3$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-2$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{3}$
$r=-1$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$