L11a72

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-2uv^{6}+3uv^{5}-4uv^{4}+4uv^{3}-3uv^{2}+2uv-u-v^{7}+2v^{6}-3v^{5}+4v^{4}-4v^{3}+3v^{2}-2v}{{\sqrt {u}}v^{7/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{7/2}}}+{\frac {2}{q^{9/2}}}-{\frac {5}{q^{11/2}}}+{\frac {7}{q^{13/2}}}-{\frac {11}{q^{15/2}}}+{\frac {11}{q^{17/2}}}-{\frac {12}{q^{19/2}}}+{\frac {11}{q^{21/2}}}-{\frac {8}{q^{23/2}}}+{\frac {5}{q^{25/2}}}-{\frac {2}{q^{27/2}}}+{\frac {1}{q^{29/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$a^{13}\left(-z^{3}\right)-4a^{13}z-3a^{13}z^{-1}+3a^{11}z^{5}+14a^{11}z^{3}+19a^{11}z+7a^{11}z^{-1}-2a^{9}z^{7}-11a^{9}z^{5}-20a^{9}z^{3}-15a^{9}z-4a^{9}z^{-1}-a^{7}z^{7}-5a^{7}z^{5}-7a^{7}z^{3}-2a^{7}z$ (db)
Kauffman polynomial	$a^{18}z^{4}-2a^{18}z^{2}+a^{18}+2a^{17}z^{5}-2a^{17}z^{3}+3a^{16}z^{6}-2a^{16}z^{4}+4a^{15}z^{7}-5a^{15}z^{5}+4a^{15}z^{3}+4a^{14}z^{8}-6a^{14}z^{6}+4a^{14}z^{4}+a^{14}z^{2}+3a^{13}z^{9}-5a^{13}z^{7}+6a^{13}z^{5}-11a^{13}z^{3}+11a^{13}z-3a^{13}z^{-1}+a^{12}z^{10}+4a^{12}z^{8}-20a^{12}z^{6}+30a^{12}z^{4}-24a^{12}z^{2}+7a^{12}+6a^{11}z^{9}-24a^{11}z^{7}+44a^{11}z^{5}-53a^{11}z^{3}+30a^{11}z-7a^{11}z^{-1}+a^{10}z^{10}+2a^{10}z^{8}-19a^{10}z^{6}+31a^{10}z^{4}-24a^{10}z^{2}+7a^{10}+3a^{9}z^{9}-14a^{9}z^{7}+26a^{9}z^{5}-29a^{9}z^{3}+17a^{9}z-4a^{9}z^{-1}+2a^{8}z^{8}-8a^{8}z^{6}+8a^{8}z^{4}-a^{8}z^{2}+a^{7}z^{7}-5a^{7}z^{5}+7a^{7}z^{3}-2a^{7}z$ (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		\

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-6

1

-8

2

1

-1

-10

3

-12

4

2

-2

-14

7

3

4

-16

5

0

-18

7

6

1

-20

4

5

1

-22

4

7

-3

-24

1

4

3

-26

1

4

-3

-28

1

-30

1

-1

Integral Khovanov Homology

(db, data source)

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