L11a419

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a419 at Knotilus!
	[edit L11a419 Quick Notes]

[edit L11a419 Further Notes and Views]

Link Presentations

[edit Notes on L11a419's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2t(1)t(3)^{2}t(2)^{2}-2t(3)^{2}t(2)^{2}+t(1)t(2)^{2}-3t(1)t(3)t(2)^{2}+4t(3)t(2)^{2}-2t(2)^{2}-4t(1)t(3)^{2}t(2)+3t(3)^{2}t(2)-3t(1)t(2)+6t(1)t(3)t(2)-6t(3)t(2)+4t(2)+2t(1)t(3)^{2}-t(3)^{2}+2t(1)-4t(1)t(3)+3t(3)-2}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$-q^{8}+4q^{7}-8q^{6}+12q^{5}-16q^{4}+18q^{3}+q^{-3}-16q^{2}-2q^{-2}+15q+6q^{-1}-9$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{-6}-z^{2}a^{-6}+z^{6}a^{-4}+2z^{4}a^{-4}+z^{2}a^{-4}-a^{-4}+z^{6}a^{-2}+2z^{4}a^{-2}+a^{2}z^{2}+3z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}+2a^{2}+4a^{-2}-2z^{4}-5z^{2}-2z^{-2}-5$ (db)
Kauffman polynomial	$z^{10}a^{-2}+z^{10}a^{-4}+2z^{9}a^{-1}+6z^{9}a^{-3}+4z^{9}a^{-5}+4z^{8}a^{-2}+9z^{8}a^{-4}+7z^{8}a^{-6}+2z^{8}+2az^{7}+z^{7}a^{-1}-7z^{7}a^{-3}+z^{7}a^{-5}+7z^{7}a^{-7}+a^{2}z^{6}-4z^{6}a^{-2}-18z^{6}a^{-4}-10z^{6}a^{-6}+4z^{6}a^{-8}+z^{6}-5az^{5}-6z^{5}a^{-1}+3z^{5}a^{-3}-9z^{5}a^{-5}-12z^{5}a^{-7}+z^{5}a^{-9}-4a^{2}z^{4}-10z^{4}a^{-2}+11z^{4}a^{-4}+4z^{4}a^{-6}-6z^{4}a^{-8}-15z^{4}+2az^{3}-z^{3}a^{-1}-4z^{3}a^{-3}+5z^{3}a^{-5}+5z^{3}a^{-7}-z^{3}a^{-9}+6a^{2}z^{2}+15z^{2}a^{-2}-z^{2}a^{-6}+z^{2}a^{-8}+19z^{2}+3az+5za^{-1}+3za^{-3}+za^{-5}-4a^{2}-8a^{-2}-2a^{-4}-9-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

17

1

-1

15

3

13

5

1

-4

11

7

3

4

9

5

-4

7

9

7

2

5

8

10

2

3

7

8

-1

1

4

10

6

-1

2

5

-3

1

5

4

-5

1

-1

-7

1

Integral Khovanov Homology

(db, data source)

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