L11a4

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a4 at Knotilus!
	[edit L11a4 Quick Notes]

[edit L11a4 Further Notes and Views]

Link Presentations

[edit Notes on L11a4's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_10,6,11,5 X₈₄₉₃ X_22,14,5,13 X_20,12,21,11 X_12,22,13,21 X_14,20,15,19 X_18,10,19,9 X_2,16,3,15
Gauss code	{1, -11, 5, -3}, {4, -1, 2, -5, 10, -4, 7, -8, 6, -9, 11, -2, 3, -10, 9, -7, 8, -6}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(2t(2)^{4}-4t(2)^{3}+5t(2)^{2}-4t(2)+2\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-q^{19/2}+4q^{17/2}-9q^{15/2}+15q^{13/2}-19q^{11/2}+22q^{9/2}-22q^{7/2}+18q^{5/2}-14q^{3/2}+7{\sqrt {q}}-{\frac {4}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-7}-2z^{3}a^{-7}-za^{-7}+z^{7}a^{-5}+3z^{5}a^{-5}+3z^{3}a^{-5}+2za^{-5}+a^{-5}z^{-1}+z^{7}a^{-3}+3z^{5}a^{-3}+2z^{3}a^{-3}-2za^{-3}-3a^{-3}z^{-1}-z^{5}a^{-1}-2z^{3}a^{-1}+za^{-1}+2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-4}-2z^{10}a^{-6}-5z^{9}a^{-3}-11z^{9}a^{-5}-6z^{9}a^{-7}-6z^{8}a^{-2}-8z^{8}a^{-4}-11z^{8}a^{-6}-9z^{8}a^{-8}-4z^{7}a^{-1}+5z^{7}a^{-3}+17z^{7}a^{-5}-8z^{7}a^{-9}+14z^{6}a^{-2}+21z^{6}a^{-4}+21z^{6}a^{-6}+11z^{6}a^{-8}-4z^{6}a^{-10}-z^{6}+11z^{5}a^{-1}+6z^{5}a^{-3}-10z^{5}a^{-5}+8z^{5}a^{-7}+12z^{5}a^{-9}-z^{5}a^{-11}-9z^{4}a^{-2}-13z^{4}a^{-4}-10z^{4}a^{-6}-3z^{4}a^{-8}+5z^{4}a^{-10}+2z^{4}-8z^{3}a^{-1}-6z^{3}a^{-3}+9z^{3}a^{-5}-6z^{3}a^{-9}+z^{3}a^{-11}+2z^{2}a^{-2}+5z^{2}a^{-4}+5z^{2}a^{-6}-2z^{2}a^{-10}-za^{-1}-3za^{-3}-4za^{-5}-2za^{-7}-3a^{-2}-3a^{-4}-a^{-6}+2a^{-1}z^{-1}+3a^{-3}z^{-1}+a^{-5}z^{-1}$ (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		\

-3

-2

-1

0

1

2

3

4

5

6

7

8

χ

20

1

18

3

-3

16

6

1

5

14

9

3

-6

12

10

6

4

10

12

9

-3

8

10

0

6

8

12

4

6

10

-4

2

3

10

7

0

1

4

-3

-2

3

-4

1

-1

Integral Khovanov Homology

(db, data source)

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