L11n63

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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_20,16,21,15 X_18,12,19,11 X_12,20,13,19 X_22,18,5,17 X_16,22,17,21 X_13,8,14,9 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, 5, -6, -9, 3, 4, -8, 7, -5, 6, -4, 8, -7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

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

-4

-3

-2

-1

0

1

2

3

4

5

χ

12

1

-1

10

2

8

5

1

-4

6

5

2

3

4

6

5

-1

2

6

5

1

0

5

7

2

-2

3

5

-2

-4

1

5

4

-6

1

3

-2

-8

2

Integral Khovanov Homology

(db, data source)

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