L11a339

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a339 at Knotilus!
	[edit L11a339 Quick Notes]

[edit L11a339 Further Notes and Views]

Link Presentations

[edit Notes on L11a339's Link Presentations]

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(t(1)^{2}t(2)^{4}-t(1)t(2)^{4}+3t(1)t(2)^{3}+t(1)^{2}t(2)^{2}-t(1)t(2)^{2}+t(2)^{2}+3t(1)t(2)-t(1)+1\right)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$14q^{9/2}-17q^{7/2}+16q^{5/2}-{\frac {1}{q^{5/2}}}-14q^{3/2}+{\frac {3}{q^{3/2}}}+q^{17/2}-3q^{15/2}+7q^{13/2}-11q^{11/2}+10{\sqrt {q}}-{\frac {7}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{9}a^{-3}+z^{7}a^{-1}-7z^{7}a^{-3}+z^{7}a^{-5}+5z^{5}a^{-1}-19z^{5}a^{-3}+5z^{5}a^{-5}+9z^{3}a^{-1}-25z^{3}a^{-3}+9z^{3}a^{-5}+8za^{-1}-15za^{-3}+7za^{-5}+2a^{-1}z^{-1}-3a^{-3}z^{-1}+a^{-5}z^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-2}-2z^{10}a^{-4}-4z^{9}a^{-1}-10z^{9}a^{-3}-6z^{9}a^{-5}-z^{8}a^{-2}-6z^{8}a^{-4}-8z^{8}a^{-6}-3z^{8}-az^{7}+13z^{7}a^{-1}+33z^{7}a^{-3}+11z^{7}a^{-5}-8z^{7}a^{-7}+20z^{6}a^{-2}+29z^{6}a^{-4}+14z^{6}a^{-6}-6z^{6}a^{-8}+11z^{6}+4az^{5}-11z^{5}a^{-1}-41z^{5}a^{-3}-11z^{5}a^{-5}+12z^{5}a^{-7}-3z^{5}a^{-9}-24z^{4}a^{-2}-33z^{4}a^{-4}-12z^{4}a^{-6}+7z^{4}a^{-8}-z^{4}a^{-10}-11z^{4}-5az^{3}+6z^{3}a^{-1}+35z^{3}a^{-3}+13z^{3}a^{-5}-9z^{3}a^{-7}+2z^{3}a^{-9}+10z^{2}a^{-2}+18z^{2}a^{-4}+6z^{2}a^{-6}-4z^{2}a^{-8}+z^{2}a^{-10}+3z^{2}+2az-7za^{-1}-17za^{-3}-6za^{-5}+2za^{-7}-3a^{-2}-3a^{-4}-a^{-6}+2a^{-1}z^{-1}+3a^{-3}z^{-1}+a^{-5}z^{-1}$ (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

18

1

-1

16

2

14

5

1

-4

12

6

2

4

10

8

5

-3

8

9

6

3

6

7

8

1

4

7

9

-2

2

5

9

4

0

2

5

-3

-2

1

5

4

-4

2

-2

-6

1

Integral Khovanov Homology

(db, data source)

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