L10n16

L10n15

L10n17

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n16 at Knotilus!

Link Presentations

[edit Notes on L10n16's Link Presentations]

Planar diagram presentation	X6172 X18,7,19,8 X4,19,1,20 X9,14,10,15 X8493 X12,5,13,6 X20,13,5,14 X11,16,12,17 X15,10,16,11 X2,18,3,17
Gauss code	{1, -10, 5, -3}, {6, -1, 2, -5, -4, 9, -8, -6, 7, 4, -9, 8, 10, -2, 3, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)\left(2t(2)^{2}-3t(2)+2\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	${\frac {9}{q^{9/2}}}-{\frac {9}{q^{7/2}}}+{\frac {5}{q^{5/2}}}-{\frac {3}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {4}{q^{17/2}}}-{\frac {6}{q^{15/2}}}+{\frac {9}{q^{13/2}}}-{\frac {10}{q^{11/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}z-3a^{7}z^{3}-5a^{7}z-2a^{7}z^{-1}+2a^{5}z^{5}+7a^{5}z^{3}+10a^{5}z+5a^{5}z^{-1}-3a^{3}z^{3}-6a^{3}z-3a^{3}z^{-1}$ (db)
Kauffman polynomial	$-z^{5}a^{11}+z^{3}a^{11}-4z^{6}a^{10}+8z^{4}a^{10}-3z^{2}a^{10}-a^{10}-5z^{7}a^{9}+9z^{5}a^{9}-3z^{3}a^{9}+2za^{9}-2z^{8}a^{8}-6z^{6}a^{8}+20z^{4}a^{8}-10z^{2}a^{8}-10z^{7}a^{7}+21z^{5}a^{7}-17z^{3}a^{7}+9za^{7}-2a^{7}z^{-1}-2z^{8}a^{6}-5z^{6}a^{6}+15z^{4}a^{6}-14z^{2}a^{6}+5a^{6}-5z^{7}a^{5}+11z^{5}a^{5}-19z^{3}a^{5}+15za^{5}-5a^{5}z^{-1}-3z^{6}a^{4}+3z^{4}a^{4}-7z^{2}a^{4}+5a^{4}-6z^{3}a^{3}+8za^{3}-3a^{3}z^{-1}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 χ -2                 3 3 -4               4 2 -2 -6             5 1   4 -8           4 4     0 -10         6 5       1 -12       3 4         1 -14     3 6           -3 -16   1 3             2 -18   3               -3 -20 1                 1

Integral Khovanov Homology (db, data source)

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

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.

L10n15 L10n17