L10n102

L10n101

L10n103

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n102 at Knotilus!

Link Presentations

[edit Notes on L10n102's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X11,18,12,19 X15,20,16,17 X19,16,20,9 X17,12,18,13 X2536 X4,9,1,10
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 $u$, $v$, $w$, ...)	$-{\frac {(x-1)\left(uvwx+uvw-uw+ux^{2}-vw+vx^{2}-x^{2}-x\right)}{{\sqrt {u}}{\sqrt {v}}{\sqrt {w}}x^{3/2}}}$ (db)
Jones polynomial	${\frac {1}{q^{9/2}}}-{\frac {1}{q^{5/2}}}-{\frac {1}{q^{23/2}}}+{\frac {1}{q^{21/2}}}-{\frac {4}{q^{19/2}}}+{\frac {2}{q^{17/2}}}-{\frac {4}{q^{15/2}}}+{\frac {1}{q^{13/2}}}-{\frac {3}{q^{11/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{13}z^{-3}-4a^{11}z^{-1}-3a^{11}z^{-3}+5za^{9}+7a^{9}z^{-1}+3a^{9}z^{-3}-z^{3}a^{7}-za^{7}-2a^{7}z^{-1}-a^{7}z^{-3}-z^{5}a^{5}-5z^{3}a^{5}-4za^{5}-a^{5}z^{-1}$ (db)
Kauffman polynomial	$-z^{7}a^{13}+6z^{5}a^{13}-13z^{3}a^{13}+13za^{13}-6a^{13}z^{-1}+a^{13}z^{-3}-z^{8}a^{12}+3z^{6}a^{12}+4z^{4}a^{12}-16z^{2}a^{12}-3a^{12}z^{-2}+13a^{12}-5z^{7}a^{11}+26z^{5}a^{11}-41z^{3}a^{11}+31za^{11}-14a^{11}z^{-1}+3a^{11}z^{-3}-z^{8}a^{10}+19z^{4}a^{10}-36z^{2}a^{10}-6a^{10}z^{-2}+24a^{10}-4z^{7}a^{9}+22z^{5}a^{9}-36z^{3}a^{9}+26za^{9}-12a^{9}z^{-1}+3a^{9}z^{-3}-3z^{6}a^{8}+16z^{4}a^{8}-21z^{2}a^{8}-3a^{8}z^{-2}+11a^{8}+z^{5}a^{7}-3z^{3}a^{7}+4za^{7}-3a^{7}z^{-1}+a^{7}z^{-3}+z^{4}a^{6}-z^{2}a^{6}-a^{6}-z^{5}a^{5}+5z^{3}a^{5}-4za^{5}+a^{5}z^{-1}$ (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   \ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                     1 1 -6                     1 1 -8               2 1     -1 -10             2         2 -12           3 6 1       2 -14         1 1 3         3 -16       1 3             2 -18     3 1               2 -20   1 4                 3 -22                       0 -24 1                     1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-6$ $i=-4$ $i=-2$ $r=-10$ ${\mathbb {Z} }$ $r=-9$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }^{4}$ ${\mathbb {Z} }^{3}$ $r=-7$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-4$ ${\mathbb {Z} }^{3}$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

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.

L10n101 L10n103