L11a548

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a548 at Knotilus!
	[edit L11a548 Quick Notes]

[edit L11a548 Further Notes and Views]

Link L11a548.

A graph, L11a548.

Link Presentations

[edit Notes on L11a548's Link Presentations]

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-2t(2)t(1)+2t(2)t(3)t(1)-2t(3)t(1)+t(2)t(4)t(1)-t(2)t(3)t(4)t(1)+2t(3)t(4)t(1)-2t(4)t(1)+2t(2)t(5)t(1)-t(2)t(3)t(5)t(1)+t(3)t(5)t(1)-2t(2)t(4)t(5)t(1)+t(2)t(3)t(4)t(5)t(1)-2t(3)t(4)t(5)t(1)+3t(4)t(5)t(1)-2t(5)t(1)+2t(1)+2t(2)-3t(2)t(3)+2t(3)-t(2)t(4)+2t(2)t(3)t(4)-2t(3)t(4)+t(4)-2t(2)t(5)+2t(2)t(3)t(5)-t(3)t(5)+2t(2)t(4)t(5)-2t(2)t(3)t(4)t(5)+2t(3)t(4)t(5)-2t(4)t(5)+t(5)-1}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}{\sqrt {t(4)}}{\sqrt {t(5)}}}}$ (db)
Jones polynomial	$q^{-9}-q^{-8}+6q^{-7}-6q^{-6}+16q^{-5}-15q^{-4}+21q^{-3}-q^{2}-15q^{-2}+5q+15q^{-1}-10$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{10}z^{-2}+a^{10}z^{-4}-7a^{8}z^{-2}-4a^{8}z^{-4}-4a^{8}+6z^{2}a^{6}+15a^{6}z^{-2}+6a^{6}z^{-4}+14a^{6}-4z^{4}a^{4}-10z^{2}a^{4}-13a^{4}z^{-2}-4a^{4}z^{-4}-16a^{4}+z^{6}a^{2}+2z^{4}a^{2}+4z^{2}a^{2}+4a^{2}z^{-2}+a^{2}z^{-4}+6a^{2}-z^{4}$ (db)
Kauffman polynomial	$z^{6}a^{10}-5z^{4}a^{10}+10z^{2}a^{10}+5a^{10}z^{-2}-a^{10}z^{-4}-10a^{10}+z^{7}a^{9}-10z^{3}a^{9}+20za^{9}-15a^{9}z^{-1}+4a^{9}z^{-3}+z^{8}a^{8}+4z^{6}a^{8}-20z^{4}a^{8}+30z^{2}a^{8}+14a^{8}z^{-2}-4a^{8}z^{-4}-25a^{8}+z^{9}a^{7}+3z^{7}a^{7}-30z^{3}a^{7}+55za^{7}-41a^{7}z^{-1}+12a^{7}z^{-3}+z^{10}a^{6}+z^{8}a^{6}+9z^{6}a^{6}-31z^{4}a^{6}+40z^{2}a^{6}+18a^{6}z^{-2}-6a^{6}z^{-4}-31a^{6}+6z^{9}a^{5}-8z^{7}a^{5}+11z^{5}a^{5}-30z^{3}a^{5}+55za^{5}-41a^{5}z^{-1}+12a^{5}z^{-3}+z^{10}a^{4}+10z^{8}a^{4}-14z^{6}a^{4}-11z^{4}a^{4}+30z^{2}a^{4}+14a^{4}z^{-2}-4a^{4}z^{-4}-25a^{4}+5z^{9}a^{3}-5z^{5}a^{3}-10z^{3}a^{3}+20za^{3}-15a^{3}z^{-1}+4a^{3}z^{-3}+10z^{8}a^{2}-15z^{6}a^{2}+10z^{2}a^{2}+5a^{2}z^{-2}-a^{2}z^{-4}-10a^{2}+10z^{7}a-15z^{5}a+5z^{6}-5z^{4}+z^{5}a^{-1}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

5

1

-1

3

4

1

6

1

-5

-1

9

4

5

-3

11

0

-5

10

4

6

-7

5

11

6

-9

11

10

1

-11

5

15

10

-13

1

0

-15

5

-17

1

0

-19

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-3$	$i=-1$
$r=-8$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-4$	${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{11}$
$r=-3$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-1$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=0$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{9}$
$r=1$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=3$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$