L11n117

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n117 at Knotilus!
	[edit L11n117 Quick Notes]

[edit L11n117 Further Notes and Views]

Link Presentations

[edit Notes on L11n117's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_17,21,18,20 X_13,19,14,18 X_19,15,20,14 X_4,21,1,22 X_10,5,11,6 X_12,3,13,4 X_22,11,5,12 X_2,9,3,10 X_8,15,9,16
Gauss code	{1, -10, 8, -6}, {7, -1, 2, -11, 10, -7, 9, -8, -4, 5, 11, -2, -3, 4, -5, 3, 6, -9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(2)^{5}+2t(1)t(2)^{4}-4t(2)^{4}-3t(1)t(2)^{3}+5t(2)^{3}+5t(1)t(2)^{2}-3t(2)^{2}-4t(1)t(2)+2t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$2{\sqrt {q}}-{\frac {5}{\sqrt {q}}}+{\frac {7}{q^{3/2}}}-{\frac {10}{q^{5/2}}}+{\frac {9}{q^{7/2}}}-{\frac {10}{q^{9/2}}}+{\frac {8}{q^{11/2}}}-{\frac {5}{q^{13/2}}}+{\frac {3}{q^{15/2}}}-{\frac {1}{q^{17/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^{3}a^{7}+za^{7}-a^{7}z^{-1}-z^{5}a^{5}-z^{3}a^{5}+3za^{5}+4a^{5}z^{-1}-2z^{5}a^{3}-7z^{3}a^{3}-9za^{3}-4a^{3}z^{-1}+2z^{3}a+3za+az^{-1}$ (db)
Kauffman polynomial	$-z^{7}a^{9}+4z^{5}a^{9}-5z^{3}a^{9}+2za^{9}-3z^{8}a^{8}+13z^{6}a^{8}-17z^{4}a^{8}+6z^{2}a^{8}+a^{8}-2z^{9}a^{7}+3z^{7}a^{7}+10z^{5}a^{7}-16z^{3}a^{7}+5za^{7}-a^{7}z^{-1}-9z^{8}a^{6}+32z^{6}a^{6}-28z^{4}a^{6}+3z^{2}a^{6}+4a^{6}-2z^{9}a^{5}-4z^{7}a^{5}+30z^{5}a^{5}-31z^{3}a^{5}+13za^{5}-4a^{5}z^{-1}-6z^{8}a^{4}+14z^{6}a^{4}-2z^{4}a^{4}-10z^{2}a^{4}+7a^{4}-8z^{7}a^{3}+23z^{5}a^{3}-25z^{3}a^{3}+14za^{3}-4a^{3}z^{-1}-5z^{6}a^{2}+9z^{4}a^{2}-10z^{2}a^{2}+4a^{2}-z^{5}a-5z^{3}a+4za-az^{-1}-3z^{2}+1$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

χ

2

-2

0

3

-2

5

3

-2

-4

5

2

3

-6

4

5

1

-8

6

5

1

-10

3

5

2

-12

2

5

-3

-14

1

3

2

-16

2

-2

-18

1

Integral Khovanov Homology

(db, data source)

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