L11a66

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)\left(v^{4}-4v^{3}+3v^{2}-4v+1\right)}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$14q^{9/2}-17q^{7/2}+16q^{5/2}-{\frac {1}{q^{5/2}}}-14q^{3/2}+{\frac {3}{q^{3/2}}}+q^{17/2}-3q^{15/2}+7q^{13/2}-11q^{11/2}+10{\sqrt {q}}-{\frac {7}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-3}-2z^{5}a^{-1}+4z^{5}a^{-3}-2z^{5}a^{-5}+az^{3}-6z^{3}a^{-1}+7z^{3}a^{-3}-6z^{3}a^{-5}+z^{3}a^{-7}+2az-5za^{-1}+7za^{-3}-6za^{-5}+2za^{-7}+az^{-1}-2a^{-1}z^{-1}+3a^{-3}z^{-1}-3a^{-5}z^{-1}+a^{-7}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-2}-z^{10}a^{-4}-3z^{9}a^{-1}-7z^{9}a^{-3}-4z^{9}a^{-5}-8z^{8}a^{-2}-12z^{8}a^{-4}-7z^{8}a^{-6}-3z^{8}-az^{7}+5z^{7}a^{-1}+10z^{7}a^{-3}-4z^{7}a^{-5}-8z^{7}a^{-7}+35z^{6}a^{-2}+37z^{6}a^{-4}+7z^{6}a^{-6}-6z^{6}a^{-8}+11z^{6}+4az^{5}+11z^{5}a^{-1}+18z^{5}a^{-3}+26z^{5}a^{-5}+12z^{5}a^{-7}-3z^{5}a^{-9}-36z^{4}a^{-2}-30z^{4}a^{-4}+2z^{4}a^{-6}+7z^{4}a^{-8}-z^{4}a^{-10}-12z^{4}-6az^{3}-23z^{3}a^{-1}-31z^{3}a^{-3}-27z^{3}a^{-5}-11z^{3}a^{-7}+2z^{3}a^{-9}+12z^{2}a^{-2}+9z^{2}a^{-4}-5z^{2}a^{-6}-5z^{2}a^{-8}+z^{2}a^{-10}+4z^{2}+4az+11za^{-1}+16za^{-3}+14za^{-5}+5za^{-7}-2a^{-2}+2a^{-6}+a^{-8}-az^{-1}-2a^{-1}z^{-1}-3a^{-3}z^{-1}-3a^{-5}z^{-1}-a^{-7}z^{-1}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

18

1

-1

16

2

14

5

1

-4

12

6

2

4

10

8

5

-3

8

9

6

3

6

7

8

1

4

7

9

-2

2

5

9

4

0

2

5

-3

-2

1

5

4

-4

2

-2

-6

1

Integral Khovanov Homology

(db, data source)

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