L11a450

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(v+w-1)(vw-v-w)(uvw-uw+u-vw+v-1)}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$-q^{8}+3q^{7}-6q^{6}+11q^{5}-15q^{4}+18q^{3}+q^{-3}-16q^{2}-3q^{-2}+16q+7q^{-1}-11$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+z^{6}a^{-4}+z^{4}a^{-2}+3z^{4}a^{-4}-z^{4}a^{-6}-2z^{4}+a^{2}z^{2}-2z^{2}a^{-2}+5z^{2}a^{-4}-2z^{2}a^{-6}-3z^{2}+a^{2}-3a^{-2}+3a^{-4}-a^{-6}-2a^{-2}z^{-2}+a^{-4}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$z^{5}a^{-9}-2z^{3}a^{-9}+3z^{6}a^{-8}-6z^{4}a^{-8}+2z^{2}a^{-8}+5z^{7}a^{-7}-10z^{5}a^{-7}+7z^{3}a^{-7}-2za^{-7}+6z^{8}a^{-6}-13z^{6}a^{-6}+15z^{4}a^{-6}-8z^{2}a^{-6}+3a^{-6}+4z^{9}a^{-5}-2z^{7}a^{-5}-9z^{5}a^{-5}+19z^{3}a^{-5}-7za^{-5}+z^{10}a^{-4}+11z^{8}a^{-4}-36z^{6}a^{-4}+52z^{4}a^{-4}-36z^{2}a^{-4}-a^{-4}z^{-2}+11a^{-4}+7z^{9}a^{-3}-9z^{7}a^{-3}-6z^{5}a^{-3}+19z^{3}a^{-3}-12za^{-3}+2a^{-3}z^{-1}+z^{10}a^{-2}+9z^{8}a^{-2}+a^{2}z^{6}-28z^{6}a^{-2}-3a^{2}z^{4}+34z^{4}a^{-2}+3a^{2}z^{2}-29z^{2}a^{-2}-2a^{-2}z^{-2}-a^{2}+11a^{-2}+3z^{9}a^{-1}+3az^{7}+z^{7}a^{-1}-8az^{5}-16z^{5}a^{-1}+6az^{3}+15z^{3}a^{-1}-az-8za^{-1}+2a^{-1}z^{-1}+4z^{8}-7z^{6}-z^{-2}+3$ (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

17

1

-1

15

2

13

4

1

-3

11

7

2

5

9

5

-4

7

9

6

3

5

7

9

2

3

9

0

1

5

10

5

-1

2

6

-4

-3

1

5

4

-5

2

-2

-7

1

Integral Khovanov Homology

(db, data source)

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