L11n72

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n72 at Knotilus!
	[edit L11n72 Quick Notes]

[edit L11n72 Further Notes and Views]

Link Presentations

[edit Notes on L11n72's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_7,16,8,17 X_22,18,5,17 X_18,12,19,11 X_12,22,13,21 X_20,14,21,13 X_14,20,15,19 X_15,8,16,9 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, 5, -6, 7, -8, -9, 3, 4, -5, 8, -7, 6, -4}

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

Polynomial invariants

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

χ

10

1

-1

8

2

6

3

1

-2

4

5

2

3

2

5

3

-2

0

5

0

-2

5

6

1

-4

3

4

-1

-6

1

5

4

-8

1

3

-2

-10

2

Integral Khovanov Homology

(db, data source)

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