L11a331

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a331 at Knotilus!
	[edit L11a331 Quick Notes]

[edit L11a331 Further Notes and Views]

Link Presentations

[edit Notes on L11a331's Link Presentations]

Planar diagram presentation	X_10,1,11,2 X_2,11,3,12 X_12,3,13,4 X_20,5,21,6 X_22,18,9,17 X_18,14,19,13 X_14,22,15,21 X_16,7,17,8 X_8,9,1,10 X_6,15,7,16 X_4,19,5,20
Gauss code	{1, -2, 3, -11, 4, -10, 8, -9}, {9, -1, 2, -3, 6, -7, 10, -8, 5, -6, 11, -4, 7, -5}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)^{3}t(2)^{5}-3t(1)^{3}t(2)^{4}+3t(1)^{2}t(2)^{4}+3t(1)^{3}t(2)^{3}-6t(1)^{2}t(2)^{3}+4t(1)t(2)^{3}+4t(1)^{2}t(2)^{2}-6t(1)t(2)^{2}+3t(2)^{2}+3t(1)t(2)-3t(2)+1}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	${\sqrt {q}}-{\frac {3}{\sqrt {q}}}+{\frac {5}{q^{3/2}}}-{\frac {8}{q^{5/2}}}+{\frac {10}{q^{7/2}}}-{\frac {12}{q^{9/2}}}+{\frac {12}{q^{11/2}}}-{\frac {11}{q^{13/2}}}+{\frac {8}{q^{15/2}}}-{\frac {6}{q^{17/2}}}+{\frac {3}{q^{19/2}}}-{\frac {1}{q^{21/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{7}z^{7}+5a^{7}z^{5}+8a^{7}z^{3}+6a^{7}z+2a^{7}z^{-1}-a^{5}z^{9}-7a^{5}z^{7}-18a^{5}z^{5}-22a^{5}z^{3}-13a^{5}z-3a^{5}z^{-1}+a^{3}z^{7}+5a^{3}z^{5}+7a^{3}z^{3}+3a^{3}z+a^{3}z^{-1}$ (db)
Kauffman polynomial	$-z^{3}a^{13}-3z^{4}a^{12}-6z^{5}a^{11}+5z^{3}a^{11}-2za^{11}-8z^{6}a^{10}+10z^{4}a^{10}-2z^{2}a^{10}-9z^{7}a^{9}+17z^{5}a^{9}-7z^{3}a^{9}+3za^{9}-8z^{8}a^{8}+18z^{6}a^{8}-6z^{4}a^{8}+z^{2}a^{8}-6z^{9}a^{7}+17z^{7}a^{7}-12z^{5}a^{7}+10z^{3}a^{7}-8za^{7}+2a^{7}z^{-1}-2z^{10}a^{6}-2z^{8}a^{6}+31z^{6}a^{6}-43z^{4}a^{6}+18z^{2}a^{6}-3a^{6}-9z^{9}a^{5}+42z^{7}a^{5}-62z^{5}a^{5}+39z^{3}a^{5}-16za^{5}+3a^{5}z^{-1}-2z^{10}a^{4}+5z^{8}a^{4}+10z^{6}a^{4}-32z^{4}a^{4}+20z^{2}a^{4}-3a^{4}-3z^{9}a^{3}+16z^{7}a^{3}-27z^{5}a^{3}+16z^{3}a^{3}-3za^{3}+a^{3}z^{-1}-z^{8}a^{2}+5z^{6}a^{2}-8z^{4}a^{2}+5z^{2}a^{2}-a^{2}$ (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

2

3

χ

2

1

-1

0

2

-2

3

1

-2

-4

5

2

3

-6

6

4

-2

-8

6

4

2

-10

6

0

-12

5

6

-1

-14

3

6

3

-16

3

5

-2

-18

3

-20

1

3

-2

-22

1

Integral Khovanov Homology

(db, data source)

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