L11n449

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n449 at Knotilus!
	[edit L11n449 Quick Notes]

[edit L11n449 Further Notes and Views]

Link Presentations

[edit Notes on L11n449's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_13,20,14,21 X_16,12,17,11 X_19,12,20,13 X_8,16,5,15 X_14,8,15,7 X_22,17,19,18 X_18,21,9,22 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 7, -6}, {-5, 3, 9, -8}, {11, -2, 4, 5, -3, -7, 6, -4, 8, -9}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(4)-1)\left(-t(1)t(4)^{2}+t(1)t(2)t(4)^{2}-2t(2)t(4)^{2}+t(4)^{2}+t(1)t(4)-t(2)t(3)t(4)+2t(1)t(3)-t(1)t(2)t(3)+t(2)t(3)-t(3)\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}t(4)^{3/2}}}$ (db)
Jones polynomial	${\sqrt {q}}-{\frac {5}{\sqrt {q}}}+{\frac {6}{q^{3/2}}}-{\frac {8}{q^{5/2}}}+{\frac {7}{q^{7/2}}}-{\frac {9}{q^{9/2}}}+{\frac {5}{q^{11/2}}}-{\frac {5}{q^{13/2}}}+{\frac {1}{q^{15/2}}}-{\frac {1}{q^{17/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{9}z^{-3}+a^{9}z^{-1}-3a^{7}z^{-3}-3a^{7}z-5a^{7}z^{-1}+3a^{5}z^{3}+3a^{5}z^{-3}+6a^{5}z+6a^{5}z^{-1}-a^{3}z^{5}-2a^{3}z^{3}-a^{3}z^{-3}-2a^{3}z-a^{3}z^{-1}+az^{3}-az-az^{-1}$ (db)
Kauffman polynomial	$a^{9}z^{7}-6a^{9}z^{5}+13a^{9}z^{3}-a^{9}z^{-3}-13a^{9}z+6a^{9}z^{-1}+a^{8}z^{8}-2a^{8}z^{6}-6a^{8}z^{4}+17a^{8}z^{2}+3a^{8}z^{-2}-13a^{8}+a^{7}z^{9}-14a^{7}z^{5}+29a^{7}z^{3}-3a^{7}z^{-3}-27a^{7}z+14a^{7}z^{-1}+5a^{6}z^{8}-14a^{6}z^{6}-3a^{6}z^{4}+28a^{6}z^{2}+6a^{6}z^{-2}-24a^{6}+a^{5}z^{9}+5a^{5}z^{7}-26a^{5}z^{5}+30a^{5}z^{3}-3a^{5}z^{-3}-22a^{5}z+12a^{5}z^{-1}+4a^{4}z^{8}-9a^{4}z^{6}+11a^{4}z^{2}+3a^{4}z^{-2}-11a^{4}+6a^{3}z^{7}-18a^{3}z^{5}+19a^{3}z^{3}-a^{3}z^{-3}-8a^{3}z+3a^{3}z^{-1}+3a^{2}z^{6}-3a^{2}z^{4}+a^{2}z^{2}+a^{2}+5az^{3}-az^{-1}+z^{2}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

χ

2

1

-1

0

4

-2

5

4

-1

-4

3

2

1

2

-6

4

5

1

-8

5

3

2

-10

4

8

4

-12

1

0

-14

4

-16

1

0

-18

1

Integral Khovanov Homology

(db, data source)

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