L11n368

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_7,14,8,15 X_15,20,16,21 X_11,19,12,18 X_17,13,18,12 X_19,22,20,17 X_21,16,22,5 X_13,8,14,9 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {-6, 5, -7, 4, -8, 7}, {10, -1, -3, 9, 11, -2, -5, 6, -9, 3, -4, 8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)+1)(t(2)-1)(t(3)-1)^{2}(t(3)+1)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$1-q^{-1}+3q^{-2}-q^{-3}+q^{-4}+2q^{-7}-2q^{-8}+2q^{-9}-q^{-10}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{10}+2a^{8}z^{2}+2a^{8}+a^{6}z^{2}+a^{6}z^{-2}+a^{6}-a^{4}z^{6}-5a^{4}z^{4}-7a^{4}z^{2}-2a^{4}z^{-2}-6a^{4}+a^{2}z^{4}+4a^{2}z^{2}+a^{2}z^{-2}+4a^{2}$ (db)
Kauffman polynomial	$a^{11}z^{7}-5a^{11}z^{5}+6a^{11}z^{3}-2a^{11}z+2a^{10}z^{8}-11a^{10}z^{6}+16a^{10}z^{4}-9a^{10}z^{2}+2a^{10}+a^{9}z^{9}-4a^{9}z^{7}-2a^{9}z^{5}+11a^{9}z^{3}-4a^{9}z+3a^{8}z^{8}-20a^{8}z^{6}+36a^{8}z^{4}-22a^{8}z^{2}+5a^{8}+a^{7}z^{9}-6a^{7}z^{7}+7a^{7}z^{5}-a^{7}z^{3}+a^{6}z^{8}-7a^{6}z^{6}+11a^{6}z^{4}-6a^{6}z^{2}+a^{6}z^{-2}-a^{6}+a^{5}z^{5}-8a^{5}z^{3}+8a^{5}z-2a^{5}z^{-1}+3a^{4}z^{6}-14a^{4}z^{4}+15a^{4}z^{2}+2a^{4}z^{-2}-8a^{4}+a^{3}z^{7}-3a^{3}z^{5}-2a^{3}z^{3}+6a^{3}z-2a^{3}z^{-1}+a^{2}z^{6}-5a^{2}z^{4}+8a^{2}z^{2}+a^{2}z^{-2}-5a^{2}$ (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

1

2

χ

1

-1

0

-3

3

1

2

-5

1

2

-7

1

2

1

0

-9

2

1

-11

2

4

2

0

-13

1

2

-15

1

2

1

0

-17

1

0

-19

1

-21

1

-1

Integral Khovanov Homology

(db, data source)

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