L10n41

L10n40

L10n42

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n41 at Knotilus!

Link Presentations

[edit Notes on L10n41's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X20,10,7,9 X2738 X4,15,5,16 X5,13,6,12 X11,16,12,17 X17,6,18,1 X14,19,15,20 X18,13,19,14
Gauss code	{1, -4, 2, -5, -6, 8}, {4, -1, 3, -2, -7, 6, 10, -9, 5, 7, -8, -10, 9, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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.

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