L11a541

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a541 at Knotilus!
	[edit L11a541 Quick Notes]

[edit L11a541 Further Notes and Views]

Link Presentations

[edit Notes on L11a541's Link Presentations]

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(3)-1)(t(4)-1)^{2}(-t(2)t(1)+t(2)t(4)t(1)-t(4)t(1)+2t(1)+t(2)-2t(2)t(4)+t(4)-1)}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}t(4)^{3/2}}}$ (db)
Jones polynomial	$-q^{7/2}+5q^{5/2}-13q^{3/2}+18{\sqrt {q}}-{\frac {25}{\sqrt {q}}}+{\frac {25}{q^{3/2}}}-{\frac {27}{q^{5/2}}}+{\frac {20}{q^{7/2}}}-{\frac {15}{q^{9/2}}}+{\frac {7}{q^{11/2}}}-{\frac {3}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}-a^{7}z^{-1}+3z^{3}a^{5}+6za^{5}+5a^{5}z^{-1}+a^{5}z^{-3}-3z^{5}a^{3}-8z^{3}a^{3}-12za^{3}-10a^{3}z^{-1}-3a^{3}z^{-3}+z^{7}a+3z^{5}a+6z^{3}a+9za+9az^{-1}+3az^{-3}-z^{5}a^{-1}-z^{3}a^{-1}-2za^{-1}-3a^{-1}z^{-1}-a^{-1}z^{-3}$ (db)
Kauffman polynomial	$-2a^{4}z^{10}-2a^{2}z^{10}-4a^{5}z^{9}-13a^{3}z^{9}-9az^{9}-4a^{6}z^{8}-12a^{4}z^{8}-24a^{2}z^{8}-16z^{8}-3a^{7}z^{7}-4a^{5}z^{7}+5a^{3}z^{7}-7az^{7}-13z^{7}a^{-1}-a^{8}z^{6}+3a^{6}z^{6}+24a^{4}z^{6}+48a^{2}z^{6}-5z^{6}a^{-2}+23z^{6}+8a^{7}z^{5}+28a^{5}z^{5}+44a^{3}z^{5}+43az^{5}+18z^{5}a^{-1}-z^{5}a^{-3}+3a^{8}z^{4}+9a^{6}z^{4}+2a^{4}z^{4}-11a^{2}z^{4}+2z^{4}a^{-2}-5z^{4}-9a^{7}z^{3}-43a^{5}z^{3}-68a^{3}z^{3}-43az^{3}-9z^{3}a^{-1}-3a^{8}z^{2}-14a^{6}z^{2}-30a^{4}z^{2}-24a^{2}z^{2}-5z^{2}+6a^{7}z+31a^{5}z+48a^{3}z+30az+7za^{-1}+a^{8}+6a^{6}+18a^{4}+21a^{2}+9-2a^{7}z^{-1}-11a^{5}z^{-1}-18a^{3}z^{-1}-14az^{-1}-5a^{-1}z^{-1}-3a^{4}z^{-2}-6a^{2}z^{-2}-3z^{-2}+a^{5}z^{-3}+3a^{3}z^{-3}+3az^{-3}+a^{-1}z^{-3}$ (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

8

1

6

4

-4

4

9

1

8

2

9

4

-5

0

16

9

7

-2

15

0

-4

12

10

2

-6

8

15

7

-8

7

12

-5

-10

2

10

8

-12

1

5

-4

-14

2

-16

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-4$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{7}$
$r=-3$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-2$	${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=-1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{15}$	${\mathbb {Z} }^{15}$
$r=0$	${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{16}$
$r=1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$