L11n40

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,7,13,8 X_4,13,1,14 X_9,18,10,19 X₃₈₄₉ X_5,14,6,15 X_15,20,16,21 X_17,22,18,5 X_21,16,22,17 X_19,10,20,11 X_11,2,12,3
Gauss code	{1, 11, -5, -3}, {-6, -1, 2, 5, -4, 10, -11, -2, 3, 6, -7, 9, -8, 4, -10, 7, -9, 8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2uv^{5}-3uv^{4}+3uv^{3}-2uv^{2}+uv+v^{4}-2v^{3}+3v^{2}-3v+2}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	${\frac {1}{q^{9/2}}}-{\frac {1}{q^{7/2}}}+{\frac {2}{q^{25/2}}}-{\frac {4}{q^{23/2}}}+{\frac {6}{q^{21/2}}}-{\frac {7}{q^{19/2}}}+{\frac {7}{q^{17/2}}}-{\frac {7}{q^{15/2}}}+{\frac {5}{q^{13/2}}}-{\frac {4}{q^{11/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$-za^{13}-a^{13}z^{-1}+z^{5}a^{11}+4z^{3}a^{11}+4za^{11}+a^{11}z^{-1}-z^{7}a^{9}-4z^{5}a^{9}-2z^{3}a^{9}+4za^{9}+2a^{9}z^{-1}-z^{7}a^{7}-6z^{5}a^{7}-12z^{3}a^{7}-9za^{7}-2a^{7}z^{-1}$ (db)
Kauffman polynomial	$-3z^{2}a^{16}+a^{16}-z^{5}a^{15}-3z^{3}a^{15}+za^{15}-3z^{6}a^{14}+3z^{4}a^{14}-2z^{2}a^{14}-4z^{7}a^{13}+8z^{5}a^{13}-4z^{3}a^{13}-za^{13}+a^{13}z^{-1}-3z^{8}a^{12}+7z^{6}a^{12}-7z^{4}a^{12}+9z^{2}a^{12}-3a^{12}-z^{9}a^{11}-z^{7}a^{11}+7z^{5}a^{11}+z^{3}a^{11}-4za^{11}+a^{11}z^{-1}-4z^{8}a^{10}+13z^{6}a^{10}-9z^{4}a^{10}+z^{2}a^{10}-z^{9}a^{9}+2z^{7}a^{9}+4z^{5}a^{9}-10z^{3}a^{9}+7za^{9}-2a^{9}z^{-1}-z^{8}a^{8}+3z^{6}a^{8}+z^{4}a^{8}-7z^{2}a^{8}+3a^{8}-z^{7}a^{7}+6z^{5}a^{7}-12z^{3}a^{7}+9za^{7}-2a^{7}z^{-1}$ (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

-6

1

-8

1

0

-10

3

-12

2

1

-1

-14

5

3

2

-16

3

0

-18

4

0

-20

2

3

1

-22

2

4

-2

-24

2

-26

2

-2

Integral Khovanov Homology

(db, data source)

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