L10a118

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Rich Schwartz' "25" [1]

Two interlaced pentagons.

Two hollow interlaced pentagrams.

Link Presentations

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Planar diagram presentation	X_12,1,13,2 X_2,13,3,14 X_14,3,15,4 X_16,5,17,6 X_18,7,19,8 X_20,9,11,10 X_10,11,1,12 X_4,15,5,16 X_6,17,7,18 X_8,19,9,20
Gauss code	{1, -2, 3, -8, 4, -9, 5, -10, 6, -7}, {7, -1, 2, -3, 8, -4, 9, -5, 10, -6}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-t(1)^{4}t(2)^{4}-t(1)^{3}t(2)^{3}-t(1)^{2}t(2)^{2}-t(1)t(2)-1}{t(1)^{2}t(2)^{2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{9/2}}}-{\frac {1}{q^{29/2}}}+{\frac {1}{q^{27/2}}}-{\frac {1}{q^{25/2}}}+{\frac {1}{q^{23/2}}}-{\frac {1}{q^{21/2}}}+{\frac {1}{q^{19/2}}}-{\frac {1}{q^{17/2}}}+{\frac {1}{q^{15/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-9 (db)
HOMFLY-PT polynomial	$a^{11}z^{7}+7a^{11}z^{5}+15a^{11}z^{3}+10a^{11}z+a^{11}z^{-1}-a^{9}z^{9}-9a^{9}z^{7}-28a^{9}z^{5}-35a^{9}z^{3}-15a^{9}z-a^{9}z^{-1}$ (db)
Kauffman polynomial	$a^{19}z+a^{18}z^{2}+a^{17}z^{3}-a^{17}z+a^{16}z^{4}-2a^{16}z^{2}+a^{15}z^{5}-3a^{15}z^{3}+a^{15}z+a^{14}z^{6}-4a^{14}z^{4}+3a^{14}z^{2}+a^{13}z^{7}-5a^{13}z^{5}+6a^{13}z^{3}-a^{13}z+a^{12}z^{8}-6a^{12}z^{6}+10a^{12}z^{4}-4a^{12}z^{2}+a^{11}z^{9}-8a^{11}z^{7}+22a^{11}z^{5}-25a^{11}z^{3}+11a^{11}z-a^{11}z^{-1}+a^{10}z^{8}-7a^{10}z^{6}+15a^{10}z^{4}-10a^{10}z^{2}+a^{10}+a^{9}z^{9}-9a^{9}z^{7}+28a^{9}z^{5}-35a^{9}z^{3}+15a^{9}z-a^{9}z^{-1}$ (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

0

-16

1

0

-18

0

-20

1

0

-22

0

-24

1

0

-26

0

-28

1

0

-30

1

Integral Khovanov Homology

(db, data source)

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