L11n82

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_20,7,21,8 X_4,21,1,22 X_5,14,6,15 X_3,10,4,11 X_11,16,12,17 X_15,12,16,13 X_13,22,14,5 X_18,9,19,10 X_17,2,18,3 X_8,19,9,20
Gauss code	{1, 10, -5, -3}, {-4, -1, 2, -11, 9, 5, -6, 7, -8, 4, -7, 6, -10, -9, 11, -2, 3, 8}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {4uv^{3}-6uv^{2}+4uv-u-v^{3}+4v^{2}-6v+4}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{5/2}}}+{\frac {2}{q^{7/2}}}-{\frac {6}{q^{9/2}}}+{\frac {8}{q^{11/2}}}-{\frac {10}{q^{13/2}}}+{\frac {10}{q^{15/2}}}-{\frac {10}{q^{17/2}}}+{\frac {7}{q^{19/2}}}-{\frac {4}{q^{21/2}}}+{\frac {2}{q^{23/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-a^{13}z^{-1}+z^{3}a^{11}+3za^{11}+2a^{11}z^{-1}-z^{5}a^{9}-2z^{3}a^{9}-za^{9}-a^{9}z^{-1}-2z^{5}a^{7}-5z^{3}a^{7}-za^{7}+a^{7}z^{-1}-z^{5}a^{5}-3z^{3}a^{5}-3za^{5}-a^{5}z^{-1}$ (db)
Kauffman polynomial	$-3z^{4}a^{14}+6z^{2}a^{14}-3a^{14}-z^{7}a^{13}-z^{5}a^{13}+3z^{3}a^{13}-3za^{13}+a^{13}z^{-1}-2z^{8}a^{12}+5z^{6}a^{12}-16z^{4}a^{12}+20z^{2}a^{12}-7a^{12}-z^{9}a^{11}-7z^{5}a^{11}+14z^{3}a^{11}-9za^{11}+2a^{11}z^{-1}-4z^{8}a^{10}+5z^{6}a^{10}-8z^{4}a^{10}+11z^{2}a^{10}-4a^{10}-z^{9}a^{9}-2z^{7}a^{9}-z^{5}a^{9}+8z^{3}a^{9}-5za^{9}+a^{9}z^{-1}-2z^{8}a^{8}-2z^{6}a^{8}+8z^{4}a^{8}-3z^{2}a^{8}-3z^{7}a^{7}+4z^{5}a^{7}-2za^{7}+a^{7}z^{-1}-2z^{6}a^{6}+3z^{4}a^{6}-a^{6}-z^{5}a^{5}+3z^{3}a^{5}-3za^{5}+a^{5}z^{-1}$ (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

-4

1

-6

2

1

-1

-8

4

-10

4

2

-2

-12

6

4

2

-14

5

0

-16

5

0

-18

2

5

3

-20

2

5

-3

-22

2

-24

2

-2

Integral Khovanov Homology

(db, data source)

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