L11n412

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n412 at Knotilus!
	[edit L11n412 Quick Notes]

[edit L11n412 Further Notes and Views]

Link Presentations

[edit Notes on L11n412's Link Presentations]

Planar diagram presentation	X₈₁₉₂ X_5,15,6,14 X_10,3,11,4 X_13,5,14,4 X₂₇₃₈ X_6,9,1,10 X_11,18,12,19 X_17,12,18,7 X_15,20,16,21 X_19,22,20,13 X_21,16,22,17
Gauss code	{1, -5, 3, 4, -2, -6}, {5, -1, 6, -3, -7, 8}, {-4, 2, -9, 11, -8, 7, -10, 9, -11, 10}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)^{2}t(3)^{3}-2t(1)^{2}t(3)^{2}+t(1)t(2)^{2}t(3)^{2}-2t(2)^{2}t(3)^{2}+t(1)t(3)^{2}+t(1)^{2}t(2)t(3)^{2}-2t(1)t(2)t(3)^{2}+t(2)t(3)^{2}+2t(1)^{2}t(3)-t(1)t(2)^{2}t(3)+2t(2)^{2}t(3)-t(1)t(3)-t(1)^{2}t(2)t(3)+2t(1)t(2)t(3)-t(2)t(3)-t(2)^{2}}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{-10}+2q^{-9}-4q^{-8}+6q^{-7}-6q^{-6}+8q^{-5}-6q^{-4}+6q^{-3}-3q^{-2}+2q^{-1}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{10}z^{-2}-a^{10}+3z^{2}a^{8}+4a^{8}z^{-2}+6a^{8}-2z^{4}a^{6}-6z^{2}a^{6}-5a^{6}z^{-2}-9a^{6}-z^{4}a^{4}+2a^{4}z^{-2}+2a^{4}+2z^{2}a^{2}+2a^{2}$ (db)
Kauffman polynomial	$a^{11}z^{7}-5a^{11}z^{5}+8a^{11}z^{3}-5a^{11}z+a^{11}z^{-1}+2a^{10}z^{8}-9a^{10}z^{6}+12a^{10}z^{4}-7a^{10}z^{2}-a^{10}z^{-2}+4a^{10}+a^{9}z^{9}+a^{9}z^{7}-20a^{9}z^{5}+35a^{9}z^{3}-21a^{9}z+5a^{9}z^{-1}+6a^{8}z^{8}-26a^{8}z^{6}+38a^{8}z^{4}-32a^{8}z^{2}-4a^{8}z^{-2}+17a^{8}+a^{7}z^{9}+4a^{7}z^{7}-28a^{7}z^{5}+44a^{7}z^{3}-33a^{7}z+9a^{7}z^{-1}+4a^{6}z^{8}-15a^{6}z^{6}+26a^{6}z^{4}-32a^{6}z^{2}-5a^{6}z^{-2}+20a^{6}+4a^{5}z^{7}-12a^{5}z^{5}+18a^{5}z^{3}-16a^{5}z+5a^{5}z^{-1}+2a^{4}z^{6}-4a^{4}z^{2}-2a^{4}z^{-2}+6a^{4}+a^{3}z^{5}+a^{3}z^{3}+a^{3}z+3a^{2}z^{2}-2a^{2}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-1

2

-3

3

2

-1

-5

3

-7

4

0

-9

4

2

-11

2

4

2

-13

4

0

-15

1

3

2

-17

1

3

-2

-19

1

-21

1

-1

Integral Khovanov Homology

(db, data source)

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