L11a490

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(3)-1)\left(-t(3)^{4}-2t(1)t(3)^{3}+t(1)t(2)t(3)^{3}-3t(2)t(3)^{3}+3t(3)^{3}+3t(1)t(3)^{2}-3t(1)t(2)t(3)^{2}+3t(2)t(3)^{2}-3t(3)^{2}-3t(1)t(3)+3t(1)t(2)t(3)-2t(2)t(3)+t(3)-t(1)t(2)\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{5/2}}}$ (db)
Jones polynomial	$-q^{8}+3q^{7}-7q^{6}+13q^{5}-18q^{4}+21q^{3}+q^{-3}-20q^{2}-3q^{-2}+19q+9q^{-1}-13$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-a^{-8}+3z^{2}a^{-6}-a^{-6}z^{-2}+2a^{-6}-3z^{4}a^{-4}-2z^{2}a^{-4}+4a^{-4}z^{-2}+4a^{-4}+z^{6}a^{-2}+a^{2}z^{2}-5z^{2}a^{-2}-5a^{-2}z^{-2}+a^{2}-9a^{-2}-2z^{4}-z^{2}+2z^{-2}+3$ (db)
Kauffman polynomial	$z^{5}a^{-9}-2z^{3}a^{-9}+za^{-9}+3z^{6}a^{-8}-5z^{4}a^{-8}+3z^{2}a^{-8}-a^{-8}+5z^{7}a^{-7}-6z^{5}a^{-7}+3z^{3}a^{-7}-2za^{-7}+a^{-7}z^{-1}+6z^{8}a^{-6}-5z^{6}a^{-6}-z^{4}a^{-6}+4z^{2}a^{-6}-a^{-6}z^{-2}+4z^{9}a^{-5}+7z^{7}a^{-5}-29z^{5}a^{-5}+37z^{3}a^{-5}-21za^{-5}+5a^{-5}z^{-1}+z^{10}a^{-4}+16z^{8}a^{-4}-38z^{6}a^{-4}+35z^{4}a^{-4}-22z^{2}a^{-4}-4a^{-4}z^{-2}+14a^{-4}+8z^{9}a^{-3}+3z^{7}a^{-3}-48z^{5}a^{-3}+68z^{3}a^{-3}-41za^{-3}+9a^{-3}z^{-1}+z^{10}a^{-2}+16z^{8}a^{-2}+a^{2}z^{6}-46z^{6}a^{-2}-3a^{2}z^{4}+52z^{4}a^{-2}+3a^{2}z^{2}-44z^{2}a^{-2}-5a^{-2}z^{-2}-a^{2}+23a^{-2}+4z^{9}a^{-1}+3az^{7}+4z^{7}a^{-1}-6az^{5}-32z^{5}a^{-1}+3az^{3}+39z^{3}a^{-1}-23za^{-1}+5a^{-1}z^{-1}+6z^{8}-15z^{6}+18z^{4}-18z^{2}-2z^{-2}+10$ (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

5

1

-4

11

8

2

6

9

11

6

-5

7

10

7

3

5

10

11

1

3

9

10

-1

1

6

12

6

-1

3

7

-4

-3

6

-5

1

3

-2

-7

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=1$	$i=3$
$r=-4$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=0$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{9}$
$r=1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=3$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=4$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{8}$
$r=5$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=6$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=7$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$