L11a273

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_2,11,3,12 X_12,3,13,4 X_14,5,15,6 X_6,9,7,10 X_18,7,19,8 X_22,19,9,20 X_20,15,21,16 X_16,21,17,22 X_8,17,1,18 X_4,13,5,14
Gauss code	{1, -2, 3, -11, 4, -5, 6, -10}, {5, -1, 2, -3, 11, -4, 8, -9, 10, -6, 7, -8, 9, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-2u^{3}v^{4}+2u^{3}v^{3}-u^{3}v^{2}-u^{2}v^{5}+3u^{2}v^{4}-5u^{2}v^{3}+3u^{2}v^{2}-u^{2}v-uv^{4}+3uv^{3}-5uv^{2}+3uv-u-v^{3}+2v^{2}-2v}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{7/2}}}+{\frac {2}{q^{9/2}}}-{\frac {5}{q^{11/2}}}+{\frac {6}{q^{13/2}}}-{\frac {9}{q^{15/2}}}+{\frac {11}{q^{17/2}}}-{\frac {11}{q^{19/2}}}+{\frac {10}{q^{21/2}}}-{\frac {8}{q^{23/2}}}+{\frac {5}{q^{25/2}}}-{\frac {3}{q^{27/2}}}+{\frac {1}{q^{29/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$a^{13}\left(-z^{3}\right)-3a^{13}z-a^{13}z^{-1}+3a^{11}z^{5}+13a^{11}z^{3}+14a^{11}z+3a^{11}z^{-1}-2a^{9}z^{7}-11a^{9}z^{5}-19a^{9}z^{3}-12a^{9}z-2a^{9}z^{-1}-a^{7}z^{7}-5a^{7}z^{5}-7a^{7}z^{3}-3a^{7}z$ (db)
Kauffman polynomial	$-z^{4}a^{18}+z^{2}a^{18}-3z^{5}a^{17}+4z^{3}a^{17}-za^{17}-4z^{6}a^{16}+4z^{4}a^{16}-4z^{7}a^{15}+3z^{5}a^{15}+2z^{3}a^{15}-2za^{15}-4z^{8}a^{14}+7z^{6}a^{14}-7z^{4}a^{14}+4z^{2}a^{14}-a^{14}-3z^{9}a^{13}+6z^{7}a^{13}-5z^{5}a^{13}+3z^{3}a^{13}-3za^{13}+a^{13}z^{-1}-z^{10}a^{12}-4z^{8}a^{12}+25z^{6}a^{12}-40z^{4}a^{12}+24z^{2}a^{12}-3a^{12}-6z^{9}a^{11}+25z^{7}a^{11}-39z^{5}a^{11}+35z^{3}a^{11}-18za^{11}+3a^{11}z^{-1}-z^{10}a^{10}-2z^{8}a^{10}+22z^{6}a^{10}-35z^{4}a^{10}+20z^{2}a^{10}-3a^{10}-3z^{9}a^{9}+14z^{7}a^{9}-23z^{5}a^{9}+23z^{3}a^{9}-13za^{9}+2a^{9}z^{-1}-2z^{8}a^{8}+8z^{6}a^{8}-7z^{4}a^{8}+z^{2}a^{8}-z^{7}a^{7}+5z^{5}a^{7}-7z^{3}a^{7}+3za^{7}$ (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

3

-2

-18

6

0

-20

4

5

1

-22

4

6

-2

-24

2

5

3

-26

1

3

-2

-28

2

-30

1

-1

Integral Khovanov Homology

(db, data source)

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