L10n48

L10n47

L10n49

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n48 at Knotilus!

Link Presentations

[edit Notes on L10n48's Link Presentations]

Planar diagram presentation	X8192 X18,9,19,10 X6718 X20,14,7,13 X12,5,13,6 X3,10,4,11 X15,5,16,4 X11,16,12,17 X14,20,15,19 X2,18,3,17
Gauss code	{1, -10, -6, 7, 5, -3}, {3, -1, 2, 6, -8, -5, 4, -9, -7, 8, 10, -2, 9, -4}

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

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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.

L10n47 L10n49