L11n13

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_5,10,6,11 X₃₈₄₉ X_11,22,12,5 X_13,20,14,21 X_19,14,20,15 X_21,12,22,13 X_9,18,10,19 X_15,2,16,3
Gauss code	{1, 11, -5, -3}, {-4, -1, 2, 5, -10, 4, -6, 9, -7, 8, -11, -2, 3, 10, -8, 7, -9, 6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {3uv^{3}-2uv^{2}-2v+3}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-{\frac {2}{q^{9/2}}}+{\frac {1}{q^{7/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{23/2}}}-{\frac {1}{q^{21/2}}}+{\frac {2}{q^{19/2}}}-{\frac {3}{q^{17/2}}}+{\frac {3}{q^{15/2}}}-{\frac {4}{q^{13/2}}}+{\frac {2}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-a^{13}z^{-1}+a^{11}z^{3}+4a^{11}z+2a^{11}z^{-1}-a^{9}z^{5}-4a^{9}z^{3}-3a^{9}z-a^{9}z^{-1}-a^{7}z^{5}-3a^{7}z^{3}+a^{7}z^{-1}-a^{5}z^{5}-4a^{5}z^{3}-3a^{5}z-a^{5}z^{-1}$ (db)
Kauffman polynomial	$-z^{4}a^{14}+4z^{2}a^{14}-3a^{14}-z^{5}a^{13}+3z^{3}a^{13}-2za^{13}+a^{13}z^{-1}-z^{8}a^{12}+7z^{6}a^{12}-19z^{4}a^{12}+21z^{2}a^{12}-7a^{12}-z^{9}a^{11}+7z^{7}a^{11}-18z^{5}a^{11}+18z^{3}a^{11}-8za^{11}+2a^{11}z^{-1}-2z^{8}a^{10}+12z^{6}a^{10}-25z^{4}a^{10}+17z^{2}a^{10}-4a^{10}-z^{9}a^{9}+6z^{7}a^{9}-13z^{5}a^{9}+11z^{3}a^{9}-6za^{9}+a^{9}z^{-1}-z^{8}a^{8}+4z^{6}a^{8}-4z^{4}a^{8}-z^{7}a^{7}+3z^{5}a^{7}-3za^{7}+a^{7}z^{-1}-z^{6}a^{6}+3z^{4}a^{6}-a^{6}-z^{5}a^{5}+4z^{3}a^{5}-3za^{5}+a^{5}z^{-1}$ (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

1

0

-8

1

-10

1

2

1

0

-12

3

1

2

-14

1

2

1

-16

2

0

-18

1

-20

1

2

-1

-22

0

-24

1

-1

Integral Khovanov Homology

(db, data source)

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