L11n253

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Link Presentations

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Planar diagram presentation	X_12,1,13,2 X_14,3,15,4 X_9,18,10,19 X_5,16,6,17 X_22,7,11,8 X_6,21,7,22 X_20,15,21,16 X_17,8,18,9 X_19,4,20,5 X_2,11,3,12 X_10,13,1,14
Gauss code	{1, -10, 2, 9, -4, -6, 5, 8, -3, -11}, {10, -1, 11, -2, 7, 4, -8, 3, -9, -7, 6, -5}

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

Polynomial invariants

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

-6

1

-8

1

0

-10

1

-12

1

-14

1

2

1

0

-16

1

0

-18

1

2

-1

-20

0

-22

1

0

-24

1

-26

1

Integral Khovanov Homology

(db, data source)

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