L11a546

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_18,16,11,15 X_20,9,21,10 X_22,13,19,14 X_14,21,15,22 X_10,19,5,20 X_8,18,9,17 X_16,8,17,7 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {7, -4, 6, -5}, {10, -1, 9, -8, 4, -7}, {11, -2, 5, -6, 3, -9, 8, -3}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {uvw^{2}x-uvw^{2}+uvwx^{2}-3uvwx+3uvw-uvx^{2}+3uvx-3uv+uw^{2}x^{2}-2uw^{2}x+uw^{2}-2uwx^{2}+4uwx-2uw+ux^{2}-2ux+u+vw^{2}x^{2}-2vw^{2}x+vw^{2}-2vwx^{2}+4vwx-2vw+vx^{2}-2vx+v-3w^{2}x^{2}+3w^{2}x-w^{2}+3wx^{2}-3wx+w-x^{2}+x}{{\sqrt {u}}{\sqrt {v}}wx}}$ (db)
Jones polynomial	$-{\frac {18}{q^{9/2}}}+{\frac {19}{q^{7/2}}}+q^{5/2}-{\frac {22}{q^{5/2}}}-4q^{3/2}+{\frac {18}{q^{3/2}}}-{\frac {1}{q^{17/2}}}+{\frac {2}{q^{15/2}}}-{\frac {8}{q^{13/2}}}+{\frac {11}{q^{11/2}}}+9{\sqrt {q}}-{\frac {15}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{9}z^{-3}+a^{9}z^{-1}-3a^{7}z^{-3}-4a^{7}z-6a^{7}z^{-1}+6a^{5}z^{3}+3a^{5}z^{-3}+12a^{5}z+9a^{5}z^{-1}-3a^{3}z^{5}-8a^{3}z^{3}-a^{3}z^{-3}-10a^{3}z-4a^{3}z^{-1}-az^{5}+az^{3}+z^{3}a^{-1}+2az$ (db)
Kauffman polynomial	$a^{9}z^{7}-5a^{9}z^{5}+10a^{9}z^{3}-a^{9}z^{-3}-10a^{9}z+5a^{9}z^{-1}+2a^{8}z^{8}-5a^{8}z^{6}+10a^{8}z^{2}+3a^{8}z^{-2}-10a^{8}+3a^{7}z^{9}-5a^{7}z^{7}-7a^{7}z^{5}+22a^{7}z^{3}-3a^{7}z^{-3}-23a^{7}z+12a^{7}z^{-1}+a^{6}z^{10}+11a^{6}z^{8}-40a^{6}z^{6}+31a^{6}z^{4}+10a^{6}z^{2}+6a^{6}z^{-2}-19a^{6}+9a^{5}z^{9}-8a^{5}z^{7}-34a^{5}z^{5}+49a^{5}z^{3}-3a^{5}z^{-3}-27a^{5}z+12a^{5}z^{-1}+a^{4}z^{10}+22a^{4}z^{8}-61a^{4}z^{6}+41a^{4}z^{4}+3a^{4}z^{-2}-10a^{4}+6a^{3}z^{9}+12a^{3}z^{7}-57a^{3}z^{5}+53a^{3}z^{3}-a^{3}z^{-3}-20a^{3}z+5a^{3}z^{-1}+13a^{2}z^{8}-17a^{2}z^{6}+2a^{2}z^{4}+z^{4}a^{-2}+2a^{2}z^{2}+14az^{7}-21az^{5}+4z^{5}a^{-1}+15az^{3}-z^{3}a^{-1}-6az+9z^{6}-7z^{4}+2z^{2}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

6

1

-1

4

3

2

6

1

-5

0

9

3

6

-2

10

7

-3

-4

12

8

4

-6

11

14

3

-8

7

8

-1

-10

6

13

7

-12

2

5

-3

-14

6

-16

1

2

-1

-18

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-8$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{2}$
$r=-6$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-5$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-4$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{7}$
$r=-3$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
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 r=-2}	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{12}$
$r=-1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=0$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{9}$
$r=1$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=3$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$