L11n252

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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_15,21,16,20 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 $u$ , $v$ , $w$ , ...)	${\frac {-t(2)^{2}t(1)^{4}-t(2)^{3}t(1)^{3}+t(2)^{2}t(1)^{3}-t(1)^{3}-t(2)^{2}t(1)^{2}-t(2)^{4}t(1)+t(2)^{2}t(1)-t(2)t(1)-t(2)^{2}}{t(1)^{2}t(2)^{2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{7/2}}}-{\frac {2}{q^{9/2}}}+{\frac {2}{q^{11/2}}}-{\frac {2}{q^{13/2}}}+{\frac {2}{q^{15/2}}}-{\frac {1}{q^{17/2}}}-{\frac {1}{q^{23/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{11}z+a^{11}z^{-1}-a^{9}z^{-1}-a^{7}z^{5}-4a^{7}z^{3}-3a^{7}z-a^{5}z^{5}-4a^{5}z^{3}-3a^{5}z$ (db)
Kauffman polynomial	$-z^{7}a^{13}+7z^{5}a^{13}-14z^{3}a^{13}+8za^{13}+z^{4}a^{12}-2z^{2}a^{12}+z^{5}a^{11}-3z^{3}a^{11}+a^{11}z^{-1}-z^{2}a^{10}-a^{10}-z^{5}a^{9}+2z^{3}a^{9}-2za^{9}+a^{9}z^{-1}-z^{6}a^{8}+2z^{4}a^{8}-z^{7}a^{7}+4z^{5}a^{7}-5z^{3}a^{7}+3za^{7}-z^{6}a^{6}+3z^{4}a^{6}-z^{2}a^{6}-z^{5}a^{5}+4z^{3}a^{5}-3za^{5}$ (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

-4

1

-6

1

0

-8

1

-10

1

0

-12

1

2

1

0

-14

1

0

-16

1

2

-1

-18

1

-20

1

0

-22

1

-24

1

Integral Khovanov Homology

(db, data source)

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