L10a151

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,4,11,3 X_14,8,5,7 X_18,14,19,13 X_16,9,17,10 X_8,15,9,16 X_20,17,15,18 X_12,20,13,19 X₂₅₃₆ X_4,12,1,11
Gauss code	{1, -9, 2, -10}, {6, -5, 7, -4, 8, -7}, {9, -1, 3, -6, 5, -2, 10, -8, 4, -3}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)(t(3)-1)\left(t(3)^{2}-3t(3)+1\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{5}-q^{-5}+3q^{4}+3q^{-4}-6q^{3}-6q^{-3}+11q^{2}+11q^{-2}-12q-12q^{-1}+14$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{6}+2a^{2}z^{4}+2z^{4}a^{-2}-3z^{4}-a^{4}z^{2}+4a^{2}z^{2}+4z^{2}a^{-2}-z^{2}a^{-4}-6z^{2}-a^{4}+4a^{2}+4a^{-2}-a^{-4}-6+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$az^{9}+z^{9}a^{-1}+3a^{2}z^{8}+3z^{8}a^{-2}+6z^{8}+4a^{3}z^{7}+9az^{7}+9z^{7}a^{-1}+4z^{7}a^{-3}+3a^{4}z^{6}+4a^{2}z^{6}+4z^{6}a^{-2}+3z^{6}a^{-4}+2z^{6}+a^{5}z^{5}-4a^{3}z^{5}-15az^{5}-15z^{5}a^{-1}-4z^{5}a^{-3}+z^{5}a^{-5}-6a^{4}z^{4}-18a^{2}z^{4}-18z^{4}a^{-2}-6z^{4}a^{-4}-24z^{4}-2a^{5}z^{3}-2a^{3}z^{3}+2az^{3}+2z^{3}a^{-1}-2z^{3}a^{-3}-2z^{3}a^{-5}+5a^{4}z^{2}+18a^{2}z^{2}+18z^{2}a^{-2}+5z^{2}a^{-4}+26z^{2}+a^{5}z+3a^{3}z+4az+4za^{-1}+3za^{-3}+za^{-5}-2a^{4}-8a^{2}-8a^{-2}-2a^{-4}-11-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

11

1

-1

9

2

7

4

1

-3

5

7

2

5

3

5

4

-1

1

9

7

2

-1

7

9

2

-3

4

5

-1

-5

2

7

5

-7

1

4

-3

-9

2

-11

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-1$	$i=1$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-2$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-1$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=0$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{9}$
$r=1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=3$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}}
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$