L11a85

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

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{5}-4uv^{4}+5uv^{3}-5uv^{2}+4uv-2u-2v^{5}+4v^{4}-5v^{3}+5v^{2}-4v+1}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {12}{q^{9/2}}}+{\frac {9}{q^{7/2}}}-{\frac {6}{q^{5/2}}}+{\frac {3}{q^{3/2}}}+{\frac {1}{q^{23/2}}}-{\frac {2}{q^{21/2}}}+{\frac {5}{q^{19/2}}}-{\frac {8}{q^{17/2}}}+{\frac {11}{q^{15/2}}}-{\frac {14}{q^{13/2}}}+{\frac {12}{q^{11/2}}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-za^{11}-2a^{11}z^{-1}+3z^{3}a^{9}+9za^{9}+5a^{9}z^{-1}-3z^{5}a^{7}-11z^{3}a^{7}-11za^{7}-3a^{7}z^{-1}+z^{7}a^{5}+4z^{5}a^{5}+5z^{3}a^{5}+3za^{5}-z^{5}a^{3}-3z^{3}a^{3}-2za^{3}$ (db)
Kauffman polynomial	$a^{14}z^{4}-2a^{14}z^{2}+a^{14}+2a^{13}z^{5}-2a^{13}z^{3}+3a^{12}z^{6}-2a^{12}z^{4}+4a^{11}z^{7}-5a^{11}z^{5}+6a^{11}z^{3}-5a^{11}z+2a^{11}z^{-1}+4a^{10}z^{8}-4a^{10}z^{6}-3a^{10}z^{4}+10a^{10}z^{2}-5a^{10}+3a^{9}z^{9}-a^{9}z^{7}-12a^{9}z^{5}+20a^{9}z^{3}-14a^{9}z+5a^{9}z^{-1}+a^{8}z^{10}+6a^{8}z^{8}-22a^{8}z^{6}+15a^{8}z^{4}+3a^{8}z^{2}-5a^{8}+6a^{7}z^{9}-14a^{7}z^{7}-a^{7}z^{5}+13a^{7}z^{3}-8a^{7}z+3a^{7}z^{-1}+a^{6}z^{10}+5a^{6}z^{8}-27a^{6}z^{6}+29a^{6}z^{4}-10a^{6}z^{2}+3a^{5}z^{9}-8a^{5}z^{7}+6a^{5}z^{3}-a^{5}z+3a^{4}z^{8}-12a^{4}z^{6}+14a^{4}z^{4}-5a^{4}z^{2}+a^{3}z^{7}-4a^{3}z^{5}+5a^{3}z^{3}-2a^{3}z$ (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

0

1

-2

2

-2

-4

4

1

3

-6

6

3

-3

-8

6

3

-10

6

0

-12

8

6

2

-14

4

7

3

-16

4

7

-3

-18

1

4

3

-20

1

4

-3

-22

1

-24

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-6$	$i=-4$
$r=-9$	${\mathbb {Z} }$
$r=-8$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-5$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-4$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{8}$
$r=-3$	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}^{6}\oplus{\mathbb Z}_2^{6}}	${\mathbb {Z} }^{6}$
$r=-2$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-1$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=0$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{4}$
$r=1$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=2$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$