L10n100

L10n99

L10n101

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n100 at Knotilus!

Link Presentations

[edit Notes on L10n100's Link Presentations]

Planar diagram presentation	X6172 X3,11,4,10 X7,15,8,14 X13,5,14,8 X11,18,12,19 X20,16,17,15 X16,20,9,19 X17,12,18,13 X2536 X9,1,10,4
Gauss code	{1, -9, -2, 10}, {9, -1, -3, 4}, {-8, 5, 7, -6}, {-10, 2, -5, 8, -4, 3, 6, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)(w-1)(x-1)}{{\sqrt {u}}{\sqrt {v}}{\sqrt {w}}{\sqrt {x}}}}$ (db)
Jones polynomial	$-2q^{9/2}+3q^{7/2}-7q^{5/2}+4q^{3/2}-7{\sqrt {q}}+{\frac {4}{\sqrt {q}}}-{\frac {4}{q^{3/2}}}+{\frac {1}{q^{5/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-a^{-5}z^{-3}-a^{-5}z^{-1}-z^{3}a^{-3}+3a^{-3}z^{-3}+za^{-3}+4a^{-3}z^{-1}+z^{5}a^{-1}-az^{3}+2z^{3}a^{-1}+az^{-3}-3a^{-1}z^{-3}-za^{-1}+2az^{-1}-5a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{7}a^{-1}-z^{7}a^{-3}-5z^{6}a^{-2}-z^{6}a^{-4}-4z^{6}-4az^{5}-6z^{5}a^{-1}-2z^{5}a^{-3}-a^{2}z^{4}+6z^{4}a^{-2}+5z^{4}+6az^{3}+10z^{3}a^{-1}+z^{3}a^{-3}-3z^{3}a^{-5}-6z^{2}a^{-2}-5z^{2}a^{-4}-z^{2}+za^{-1}+5za^{-3}+4za^{-5}+11a^{-2}+6a^{-4}+6-3az^{-1}-6a^{-1}z^{-1}-6a^{-3}z^{-1}-3a^{-5}z^{-1}-6a^{-2}z^{-2}-3a^{-4}z^{-2}-3z^{-2}+az^{-3}+3a^{-1}z^{-3}+3a^{-3}z^{-3}+a^{-5}z^{-3}$ (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 χ 10               2 2 8             3 2 -1 6           4     4 4           3     3 2       7 4       3 0     3 6         3 -2   1 1           0 -4   3             3 -6 1               -1

Integral Khovanov Homology (db, data source)

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

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

L10n99 L10n101