L11n414

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n414 at Knotilus!
	[edit L11n414 Quick Notes]

[edit L11n414 Further Notes and Views]

Link Presentations

[edit Notes on L11n414'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_18,12,19,11 X_12,18,7,17 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

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {-t(1)t(3)^{3}+t(1)t(2)t(3)^{3}-t(2)t(3)^{3}+t(1)^{2}t(3)^{2}+t(2)^{2}t(3)^{2}-t(1)^{2}t(3)-t(2)^{2}t(3)+t(1)t(2)^{2}+t(1)^{2}t(2)-t(1)t(2)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{3}-q^{2}+2q-1+2q^{-1}+q^{-3}+q^{-4}-q^{-5}+q^{-6}-q^{-7}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{6}\left(-z^{2}\right)-a^{6}z^{-2}-2a^{6}+a^{4}z^{4}+5a^{4}z^{2}+4a^{4}z^{-2}+8a^{4}-a^{2}z^{4}-5a^{2}z^{2}-5a^{2}z^{-2}+z^{2}a^{-2}-8a^{2}+2a^{-2}-z^{4}-3z^{2}+2z^{-2}$ (db)
Kauffman polynomial	$a^{7}z^{7}-6a^{7}z^{5}+10a^{7}z^{3}-5a^{7}z+a^{7}z^{-1}+a^{6}z^{8}-6a^{6}z^{6}+10a^{6}z^{4}-7a^{6}z^{2}-a^{6}z^{-2}+4a^{6}+2a^{5}z^{7}-14a^{5}z^{5}+28a^{5}z^{3}-21a^{5}z+5a^{5}z^{-1}+a^{4}z^{8}-8a^{4}z^{6}+20a^{4}z^{4}-25a^{4}z^{2}-4a^{4}z^{-2}+17a^{4}+2a^{3}z^{7}-16a^{3}z^{5}+37a^{3}z^{3}-33a^{3}z+9a^{3}z^{-1}+a^{2}z^{8}-8a^{2}z^{6}+z^{6}a^{-2}+22a^{2}z^{4}-5z^{4}a^{-2}-31a^{2}z^{2}+6z^{2}a^{-2}-5a^{2}z^{-2}+20a^{2}-2a^{-2}+2az^{7}+z^{7}a^{-1}-12az^{5}-4z^{5}a^{-1}+21az^{3}+2z^{3}a^{-1}-16az+za^{-1}+5az^{-1}+z^{8}-5z^{6}+7z^{4}-7z^{2}-2z^{-2}+6$ (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

7

1

5

1

0

3

1

-1

1

3

1

-3

1

2

-5

1

4

2

1

-7

1

2

-9

1

0

-11

1

0

-13

0

-15

1

-1

Integral Khovanov Homology

(db, data source)

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