L11a498

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(3)-1)\left(t(2)t(3)^{4}+2t(1)t(2)t(3)^{3}-3t(2)t(3)^{3}+2t(3)^{3}+4t(1)t(3)^{2}-4t(1)t(2)t(3)^{2}+4t(2)t(3)^{2}-4t(3)^{2}-3t(1)t(3)+2t(1)t(2)t(3)+2t(3)+t(1)\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{5/2}}}$ (db)
Jones polynomial	$q^{4}-4q^{3}+10q^{2}-14q+19-20q^{-1}+21q^{-2}-16q^{-3}+12q^{-4}-7q^{-5}+3q^{-6}-q^{-7}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}\left(-z^{2}\right)-a^{6}z^{-2}-2a^{6}+3a^{4}z^{4}+9a^{4}z^{2}+4a^{4}z^{-2}+10a^{4}-2a^{2}z^{6}-8a^{2}z^{4}+z^{4}a^{-2}-15a^{2}z^{2}-5a^{2}z^{-2}+z^{2}a^{-2}-13a^{2}+a^{-2}-z^{6}-z^{4}+2z^{2}+2z^{-2}+4$ (db)
Kauffman polynomial	$a^{4}z^{10}+a^{2}z^{10}+3a^{5}z^{9}+9a^{3}z^{9}+6az^{9}+3a^{6}z^{8}+12a^{4}z^{8}+22a^{2}z^{8}+13z^{8}+a^{7}z^{7}-3a^{5}z^{7}-8a^{3}z^{7}+10az^{7}+14z^{7}a^{-1}-11a^{6}z^{6}-49a^{4}z^{6}-66a^{2}z^{6}+10z^{6}a^{-2}-18z^{6}-4a^{7}z^{5}-17a^{5}z^{5}-39a^{3}z^{5}-48az^{5}-18z^{5}a^{-1}+4z^{5}a^{-3}+14a^{6}z^{4}+60a^{4}z^{4}+66a^{2}z^{4}-10z^{4}a^{-2}+z^{4}a^{-4}+9z^{4}+6a^{7}z^{3}+34a^{5}z^{3}+66a^{3}z^{3}+44az^{3}+6z^{3}a^{-1}-8a^{6}z^{2}-35a^{4}z^{2}-43a^{2}z^{2}+6z^{2}a^{-2}-10z^{2}-4a^{7}z-21a^{5}z-39a^{3}z-22az+3a^{6}+16a^{4}+21a^{2}-2a^{-2}+7+a^{7}z^{-1}+5a^{5}z^{-1}+9a^{3}z^{-1}+5az^{-1}-a^{6}z^{-2}-4a^{4}z^{-2}-5a^{2}z^{-2}-2z^{-2}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

9

1

7

4

1

-3

5

6

3

8

4

-4

1

11

6

5

-1

11

10

-1

-3

10

9

1

-5

6

11

5

-7

6

10

-4

-9

2

7

5

-11

1

5

-4

-13

2

-15

1

-1

Integral Khovanov Homology

(db, data source)

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