L11a33

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-2uv^{6}+3uv^{5}-3uv^{4}+3uv^{3}-3uv^{2}+2uv-u-v^{7}+2v^{6}-3v^{5}+3v^{4}-3v^{3}+3v^{2}-2v}{{\sqrt {u}}v^{7/2}}}$ (db)
Jones polynomial	${\frac {2}{q^{9/2}}}-{\frac {1}{q^{7/2}}}+{\frac {1}{q^{29/2}}}-{\frac {2}{q^{27/2}}}+{\frac {5}{q^{25/2}}}-{\frac {7}{q^{23/2}}}+{\frac {9}{q^{21/2}}}-{\frac {11}{q^{19/2}}}+{\frac {10}{q^{17/2}}}-{\frac {9}{q^{15/2}}}+{\frac {6}{q^{13/2}}}-{\frac {5}{q^{11/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$-z^{3}a^{13}-4za^{13}-3a^{13}z^{-1}+3z^{5}a^{11}+14z^{3}a^{11}+18za^{11}+7a^{11}z^{-1}-2z^{7}a^{9}-11z^{5}a^{9}-19z^{3}a^{9}-13za^{9}-4a^{9}z^{-1}-z^{7}a^{7}-5z^{5}a^{7}-7z^{3}a^{7}-3za^{7}$ (db)
Kauffman polynomial	$-z^{4}a^{18}+2z^{2}a^{18}-a^{18}-2z^{5}a^{17}+2z^{3}a^{17}-3z^{6}a^{16}+3z^{4}a^{16}-z^{2}a^{16}-3z^{7}a^{15}+2z^{5}a^{15}-3z^{8}a^{14}+4z^{6}a^{14}-3z^{4}a^{14}+z^{2}a^{14}-3z^{9}a^{13}+9z^{7}a^{13}-17z^{5}a^{13}+20z^{3}a^{13}-12za^{13}+3a^{13}z^{-1}-z^{10}a^{12}-3z^{8}a^{12}+22z^{6}a^{12}-39z^{4}a^{12}+29z^{2}a^{12}-7a^{12}-6z^{9}a^{11}+27z^{7}a^{11}-49z^{5}a^{11}+53z^{3}a^{11}-31za^{11}+7a^{11}z^{-1}-z^{10}a^{10}-2z^{8}a^{10}+23z^{6}a^{10}-39z^{4}a^{10}+26z^{2}a^{10}-7a^{10}-3z^{9}a^{9}+14z^{7}a^{9}-23z^{5}a^{9}+24z^{3}a^{9}-16za^{9}+4a^{9}z^{-1}-2z^{8}a^{8}+8z^{6}a^{8}-7z^{4}a^{8}+z^{2}a^{8}-z^{7}a^{7}+5z^{5}a^{7}-7z^{3}a^{7}+3za^{7}$ (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

3

2

-1

-14

6

3

-16

5

4

-1

-18

6

5

1

-20

3

5

2

-22

4

6

-2

-24

1

3

2

-26

1

4

-3

-28

1

-30

1

-1

Integral Khovanov Homology

(db, data source)

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