L11a468

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(3)-1)\left(t(2)^{2}t(3)^{3}-t(2)t(3)^{3}+2t(1)t(2)^{2}t(3)^{2}-2t(2)^{2}t(3)^{2}-2t(1)t(2)t(3)^{2}+2t(2)t(3)^{2}-t(3)^{2}-t(1)t(2)^{2}t(3)-2t(1)t(3)+2t(1)t(2)t(3)-2t(2)t(3)+2t(3)+t(1)-t(1)t(2)\right)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$1-2q^{-1}+5q^{-2}-8q^{-3}+12q^{-4}-13q^{-5}+15q^{-6}-12q^{-7}+10q^{-8}-6q^{-9}+3q^{-10}-q^{-11}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{10}\left(-z^{2}\right)-2a^{10}+3a^{8}z^{4}+9a^{8}z^{2}+a^{8}z^{-2}+6a^{8}-2a^{6}z^{6}-8a^{6}z^{4}-11a^{6}z^{2}-2a^{6}z^{-2}-8a^{6}-a^{4}z^{6}-2a^{4}z^{4}+2a^{4}z^{2}+a^{4}z^{-2}+3a^{4}+a^{2}z^{4}+3a^{2}z^{2}+a^{2}$ (db)
Kauffman polynomial	$z^{5}a^{13}-2z^{3}a^{13}+za^{13}+3z^{6}a^{12}-6z^{4}a^{12}+3z^{2}a^{12}-a^{12}+4z^{7}a^{11}-5z^{5}a^{11}-2z^{3}a^{11}+za^{11}+4z^{8}a^{10}-4z^{6}a^{10}-2z^{4}a^{10}+z^{2}a^{10}+3z^{9}a^{9}-2z^{7}a^{9}-4z^{5}a^{9}+8z^{3}a^{9}-3za^{9}+z^{10}a^{8}+6z^{8}a^{8}-25z^{6}a^{8}+42z^{4}a^{8}-29z^{2}a^{8}-a^{8}z^{-2}+10a^{8}+6z^{9}a^{7}-16z^{7}a^{7}+14z^{5}a^{7}+7z^{3}a^{7}-10za^{7}+2a^{7}z^{-1}+z^{10}a^{6}+5z^{8}a^{6}-28z^{6}a^{6}+51z^{4}a^{6}-40z^{2}a^{6}-2a^{6}z^{-2}+14a^{6}+3z^{9}a^{5}-8z^{7}a^{5}+6z^{5}a^{5}+2z^{3}a^{5}-7za^{5}+2a^{5}z^{-1}+3z^{8}a^{4}-9z^{6}a^{4}+9z^{4}a^{4}-9z^{2}a^{4}-a^{4}z^{-2}+5a^{4}+2z^{7}a^{3}-6z^{5}a^{3}+3z^{3}a^{3}+z^{6}a^{2}-4z^{4}a^{2}+4z^{2}a^{2}-a^{2}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

1

-1

1

-1

-3

4

1

3

-5

5

2

-3

-7

7

3

4

-9

7

6

-1

-11

8

6

2

-13

5

8

3

-15

5

7

-2

-17

2

6

4

-19

1

4

-3

-21

2

-23

1

-1

Integral Khovanov Homology

(db, data source)

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