L10n29

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n29 at Knotilus!
	[edit L10n29 Quick Notes]

[edit L10n29 Further Notes and Views]

Link Presentations

[edit Notes on L10n29's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_3,10,4,11 X_7,14,8,15 X_15,20,16,5 X_9,17,10,16 X_19,9,20,8 X_13,19,14,18 X_17,13,18,12 X₂₅₃₆ X_11,4,12,1
Gauss code	{1, -9, -2, 10}, {9, -1, -3, 6, -5, 2, -10, 8, -7, 3, -4, 5, -8, 7, -6, 4}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(2)^{5}-3t(1)t(2)^{4}+t(2)^{4}+4t(1)t(2)^{3}-4t(2)^{3}-4t(1)t(2)^{2}+4t(2)^{2}+t(1)t(2)-3t(2)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-q^{7/2}+3q^{5/2}-6q^{3/2}+8{\sqrt {q}}-{\frac {9}{\sqrt {q}}}+{\frac {9}{q^{3/2}}}-{\frac {8}{q^{5/2}}}+{\frac {5}{q^{7/2}}}-{\frac {3}{q^{9/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-a^{3}z^{5}+5az^{5}-z^{5}a^{-1}-4a^{3}z^{3}+9az^{3}-3z^{3}a^{-1}+a^{5}z-7a^{3}z+7az-3za^{-1}+2a^{5}z^{-1}-4a^{3}z^{-1}+3az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2a^{2}z^{8}-2z^{8}-5a^{3}z^{7}-9az^{7}-4z^{7}a^{-1}-3a^{4}z^{6}-3a^{2}z^{6}-3z^{6}a^{-2}-3z^{6}+13a^{3}z^{5}+21az^{5}+7z^{5}a^{-1}-z^{5}a^{-3}+3a^{4}z^{4}+11a^{2}z^{4}+6z^{4}a^{-2}+14z^{4}-6a^{5}z^{3}-23a^{3}z^{3}-21az^{3}-2z^{3}a^{-1}+2z^{3}a^{-3}-4a^{4}z^{2}-11a^{2}z^{2}-3z^{2}a^{-2}-10z^{2}+8a^{5}z+17a^{3}z+12az+2za^{-1}-za^{-3}+2a^{4}+3a^{2}+a^{-2}+3-2a^{5}z^{-1}-4a^{3}z^{-1}-3az^{-1}-a^{-1}z^{-1}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

2

-2

4

1

3

2

4

2

-2

0

5

4

1

-2

5

0

-4

3

4

-1

-6

2

5

3

-8

1

3

-2

-10

3

Integral Khovanov Homology

(db, data source)

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