L11a489

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a489 at Knotilus!
	[edit L11a489 Quick Notes]

[edit L11a489 Further Notes and Views]

Link Presentations

[edit Notes on L11a489's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(3)-1)\left(t(2)t(3)^{4}+2t(1)t(2)t(3)^{3}-3t(2)t(3)^{3}+2t(3)^{3}+3t(1)t(3)^{2}-3t(1)t(2)t(3)^{2}+3t(2)t(3)^{2}-3t(3)^{2}-3t(1)t(3)+2t(1)t(2)t(3)+2t(3)+t(1)\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{5/2}}}$ (db)
Jones polynomial	$q^{4}-3q^{3}+7q^{2}-11q+17-17q^{-1}+18q^{-2}-15q^{-3}+12q^{-4}-7q^{-5}+3q^{-6}-q^{-7}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{2}a^{6}-a^{6}z^{-2}-2a^{6}+3z^{4}a^{4}+9z^{2}a^{4}+4a^{4}z^{-2}+10a^{4}-2z^{6}a^{2}-8z^{4}a^{2}-14z^{2}a^{2}-5a^{2}z^{-2}-13a^{2}-z^{6}-2z^{4}+2z^{-2}+4+z^{4}a^{-2}+2z^{2}a^{-2}+a^{-2}$ (db)
Kauffman polynomial	$a^{4}z^{10}+a^{2}z^{10}+3a^{5}z^{9}+8a^{3}z^{9}+5az^{9}+3a^{6}z^{8}+11a^{4}z^{8}+17a^{2}z^{8}+9z^{8}+a^{7}z^{7}-3a^{5}z^{7}-8a^{3}z^{7}+4az^{7}+8z^{7}a^{-1}-11a^{6}z^{6}-47a^{4}z^{6}-57a^{2}z^{6}+6z^{6}a^{-2}-15z^{6}-4a^{7}z^{5}-18a^{5}z^{5}-37a^{3}z^{5}-35az^{5}-9z^{5}a^{-1}+3z^{5}a^{-3}+14a^{6}z^{4}+57a^{4}z^{4}+63a^{2}z^{4}-7z^{4}a^{-2}+z^{4}a^{-4}+12z^{4}+6a^{7}z^{3}+34a^{5}z^{3}+64a^{3}z^{3}+42az^{3}+4z^{3}a^{-1}-2z^{3}a^{-3}-8a^{6}z^{2}-36a^{4}z^{2}-43a^{2}z^{2}+6z^{2}a^{-2}-z^{2}a^{-4}-8z^{2}-4a^{7}z-21a^{5}z-39a^{3}z-22az+3a^{6}+16a^{4}+21a^{2}-2a^{-2}+7+a^{7}z^{-1}+5a^{5}z^{-1}+9a^{3}z^{-1}+5az^{-1}-a^{6}z^{-2}-4a^{4}z^{-2}-5a^{2}z^{-2}-2z^{-2}$ (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

3

1

-2

5

4

3

7

3

-4

1

10

4

6

-1

9

0

-3

9

8

1

-5

6

9

3

-7

6

9

-3

-9

2

7

5

-11

1

5

-4

-13

2

-15

1

-1

Integral Khovanov Homology

(db, data source)

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