L11n92

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n92 at Knotilus!
	[edit L11n92 Quick Notes]

[edit L11n92 Further Notes and Views]

Link Presentations

[edit Notes on L11n92's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_7,16,8,17 X_20,12,21,11 X_22,18,5,17 X_18,22,19,21 X_10,20,11,19 X_9,14,10,15 X_15,8,16,9 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 9, -8, -7, 4, -2, 11, 8, -9, 3, 5, -6, 7, -4, 6, -5}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2(t(1)-1)(t(2)-1)^{3}}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-7q^{9/2}+9q^{7/2}-11q^{5/2}+{\frac {1}{q^{5/2}}}+11q^{3/2}-{\frac {4}{q^{3/2}}}-q^{13/2}+4q^{11/2}-10{\sqrt {q}}+{\frac {6}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{-5}+z^{5}a^{-3}+z^{3}a^{-3}+z^{5}a^{-1}-az^{3}+z^{3}a^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{9}a^{-1}-z^{9}a^{-3}-5z^{8}a^{-2}-4z^{8}a^{-4}-z^{8}-6z^{7}a^{-3}-6z^{7}a^{-5}+8z^{6}a^{-2}+4z^{6}a^{-4}-4z^{6}a^{-6}-4az^{5}-4z^{5}a^{-1}+13z^{5}a^{-3}+12z^{5}a^{-5}-z^{5}a^{-7}-a^{2}z^{4}-6z^{4}a^{-2}+3z^{4}a^{-4}+7z^{4}a^{-6}-3z^{4}+5az^{3}+4z^{3}a^{-1}-6z^{3}a^{-3}-4z^{3}a^{-5}+z^{3}a^{-7}+a^{2}z^{2}-2z^{2}a^{-4}-z^{2}a^{-6}+2z^{2}+1-az^{-1}-a^{-1}z^{-1}$ (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		\

-3

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

3

-3

10

4

1

3

8

5

3

-2

6

4

2

4

5

0

2

5

6

-1

0

3

7

4

-2

1

3

-2

-4

3

-6

1

-1

Integral Khovanov Homology

(db, data source)

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