L11a540

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(x-1)\left(uvwx^{2}-2uvwx+2uvx-uv-uwx^{2}+3uwx-2ux+2u-2vwx^{2}+2vwx-3vx+v+wx^{2}-2wx+2x-1\right)}{{\sqrt {u}}{\sqrt {v}}{\sqrt {w}}x^{3/2}}}$ (db)
Jones polynomial	$q^{3/2}-4{\sqrt {q}}+{\frac {8}{\sqrt {q}}}-{\frac {14}{q^{3/2}}}+{\frac {15}{q^{5/2}}}-{\frac {19}{q^{7/2}}}+{\frac {17}{q^{9/2}}}-{\frac {16}{q^{11/2}}}+{\frac {9}{q^{13/2}}}-{\frac {6}{q^{15/2}}}+{\frac {2}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}z^{-3}+a^{9}z+2a^{9}z^{-1}-3a^{7}z^{3}-3a^{7}z^{-3}-9a^{7}z-9a^{7}z^{-1}+3a^{5}z^{5}+11a^{5}z^{3}+3a^{5}z^{-3}+17a^{5}z+12a^{5}z^{-1}-a^{3}z^{7}-4a^{3}z^{5}-8a^{3}z^{3}-a^{3}z^{-3}-10a^{3}z-5a^{3}z^{-1}+az^{5}+2az^{3}+az$ (db)
Kauffman polynomial	$-z^{5}a^{11}+3z^{3}a^{11}-3za^{11}+a^{11}z^{-1}-2z^{6}a^{10}+3z^{4}a^{10}-a^{10}-3z^{7}a^{9}+3z^{5}a^{9}-z^{3}a^{9}+3za^{9}-3a^{9}z^{-1}+a^{9}z^{-3}-3z^{8}a^{8}-2z^{6}a^{8}+12z^{4}a^{8}-16z^{2}a^{8}-3a^{8}z^{-2}+11a^{8}-3z^{9}a^{7}-z^{7}a^{7}+9z^{5}a^{7}-20z^{3}a^{7}+20za^{7}-12a^{7}z^{-1}+3a^{7}z^{-3}-z^{10}a^{6}-9z^{8}a^{6}+19z^{6}a^{6}-3z^{4}a^{6}-28z^{2}a^{6}-6a^{6}z^{-2}+24a^{6}-7z^{9}a^{5}+4z^{7}a^{5}+23z^{5}a^{5}-39z^{3}a^{5}+29za^{5}-14a^{5}z^{-1}+3a^{5}z^{-3}-z^{10}a^{4}-12z^{8}a^{4}+32z^{6}a^{4}-16z^{4}a^{4}-12z^{2}a^{4}-3a^{4}z^{-2}+13a^{4}-4z^{9}a^{3}-2z^{7}a^{3}+28z^{5}a^{3}-31z^{3}a^{3}+18za^{3}-6a^{3}z^{-1}+a^{3}z^{-3}-6z^{8}a^{2}+12z^{6}a^{2}-2z^{4}a^{2}-z^{2}a^{2}-4z^{7}a+10z^{5}a-8z^{3}a+3za-z^{6}+2z^{4}-z^{2}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

4

1

-1

2

3

0

5

1

-4

-2

9

3

6

-4

9

8

-1

-6

10

6

4

-8

7

9

2

-10

9

10

-1

-12

4

11

7

-14

2

5

-3

-16

4

-18

1

2

-1

-20

1

Integral Khovanov Homology

(db, data source)

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