L11n127

L11n126

L11n128

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n127 at Knotilus!

Link Presentations

[edit Notes on L11n127's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X22,10,7,9 X2738 X4,15,5,16 X5,13,6,12 X11,16,12,17 X17,6,18,1 X14,20,15,19 X20,14,21,13 X18,21,19,22
Gauss code	{1, -4, 2, -5, -6, 8}, {4, -1, 3, -2, -7, 6, 10, -9, 5, 7, -8, -11, 9, -10, 11, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{2}v^{2}-2u^{2}v+u^{2}-uv^{2}+uv-u+v^{2}-2v+1}{uv}}$ (db)
Jones polynomial	$-q^{5/2}+2q^{3/2}-3{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {4}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {3}{q^{7/2}}}+{\frac {2}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^{5}z+a^{5}z^{-1}-2a^{3}z^{3}-5a^{3}z-2a^{3}z^{-1}+az^{5}+4az^{3}-z^{3}a^{-1}+5az+2az^{-1}-2za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{9}-az^{9}-2a^{4}z^{8}-4a^{2}z^{8}-2z^{8}-a^{5}z^{7}+2a^{3}z^{7}+2az^{7}-z^{7}a^{-1}+10a^{4}z^{6}+20a^{2}z^{6}+10z^{6}+5a^{5}z^{5}+9a^{3}z^{5}+9az^{5}+5z^{5}a^{-1}-13a^{4}z^{4}-26a^{2}z^{4}-13z^{4}-7a^{5}z^{3}-20a^{3}z^{3}-20az^{3}-7z^{3}a^{-1}+5a^{4}z^{2}+10a^{2}z^{2}+5z^{2}+4a^{5}z+12a^{3}z+12az+4za^{-1}-a^{2}-a^{5}z^{-1}-2a^{3}z^{-1}-2az^{-1}-a^{-1}z^{-1}$ (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 6                 1 1 4               1   -1 2             2 1   1 0           2 2     0 -2         2 2 1     1 -4       2 3         1 -6     1 2 1         0 -8   1 2             1 -10   1               -1 -12 1                 1

Integral Khovanov Homology (db, data source)

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

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.

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