L11n222

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_2,11,3,12 X_12,3,13,4 X_13,20,14,21 X_5,14,6,15 X_4,21,5,22 X_16,9,17,10 X_22,15,9,16 X_17,6,18,7 X_7,18,8,19 X_19,8,20,1
Gauss code	{1, -2, 3, -6, -5, 9, -10, 11}, {7, -1, 2, -3, -4, 5, 8, -7, -9, 10, -11, 4, 6, -8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(uv+1)\left(u^{2}v^{4}-uv^{3}+2uv^{2}-uv+1\right)}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{9/2}}}-{\frac {1}{q^{27/2}}}+{\frac {2}{q^{25/2}}}-{\frac {2}{q^{23/2}}}+{\frac {2}{q^{21/2}}}-{\frac {2}{q^{19/2}}}+{\frac {1}{q^{17/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$a^{13}\left(-z^{3}\right)-2a^{13}z-a^{13}z^{-1}+a^{11}z^{7}+8a^{11}z^{5}+19a^{11}z^{3}+15a^{11}z+3a^{11}z^{-1}-a^{9}z^{9}-9a^{9}z^{7}-28a^{9}z^{5}-36a^{9}z^{3}-17a^{9}z-2a^{9}z^{-1}$ (db)
Kauffman polynomial	$-z^{3}a^{17}+za^{17}-2z^{4}a^{16}+3z^{2}a^{16}-z^{5}a^{15}+za^{15}-2z^{4}a^{14}+2z^{2}a^{14}-a^{14}-z^{3}a^{13}-za^{13}+a^{13}z^{-1}-z^{8}a^{12}+8z^{6}a^{12}-19z^{4}a^{12}+14z^{2}a^{12}-3a^{12}-z^{9}a^{11}+9z^{7}a^{11}-27z^{5}a^{11}+34z^{3}a^{11}-18za^{11}+3a^{11}z^{-1}-z^{8}a^{10}+8z^{6}a^{10}-19z^{4}a^{10}+15z^{2}a^{10}-3a^{10}-z^{9}a^{9}+9z^{7}a^{9}-28z^{5}a^{9}+36z^{3}a^{9}-17za^{9}+2a^{9}z^{-1}$ (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		\

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-8

1

-10

1

-12

1

-14

1

-16

1

-1

-18

2

1

-20

1

0

-22

2

0

-24

1

0

-26

1

-1

-28

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-10$	$i=-8$	$i=-6$
$r=-10$		${\mathbb {Z} }$
$r=-9$		${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-8$		${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{2}$
$r=-7$		${\mathbb {Z} }^{2}$
$r=-6$		${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-5$	${\mathbb {Z} }$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$		${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$