L11n143

L11n142

L11n144

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n143 at Knotilus!

Link Presentations

[edit Notes on L11n143's Link Presentations]

Planar diagram presentation	X8192 X11,19,12,18 X3,10,4,11 X17,3,18,2 X12,5,13,6 X6718 X16,10,17,9 X20,16,21,15 X22,14,7,13 X14,22,15,21 X4,20,5,19
Gauss code	{1, 4, -3, -11, 5, -6}, {6, -1, 7, 3, -2, -5, 9, -10, 8, -7, -4, 2, 11, -8, 10, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2u^{2}v^{2}-2u^{2}v+uv-2v+2}{uv}}$ (db)
Jones polynomial	$q^{13/2}-q^{11/2}+2q^{9/2}-3q^{7/2}+2q^{5/2}-3q^{3/2}+2{\sqrt {q}}-{\frac {2}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^{3}a^{-5}+3za^{-5}+a^{-5}z^{-1}-z^{5}a^{-3}-4z^{3}a^{-3}-4za^{-3}-2a^{-3}z^{-1}-z^{5}a^{-1}+az^{3}-4z^{3}a^{-1}+3az-3za^{-1}+az^{-1}$ (db)
Kauffman polynomial	$z^{8}a^{-6}-7z^{6}a^{-6}+16z^{4}a^{-6}-13z^{2}a^{-6}+2a^{-6}+z^{9}a^{-5}-6z^{7}a^{-5}+11z^{5}a^{-5}-8z^{3}a^{-5}+4za^{-5}-a^{-5}z^{-1}+3z^{8}a^{-4}-18z^{6}a^{-4}+33z^{4}a^{-4}-22z^{2}a^{-4}+5a^{-4}+z^{9}a^{-3}-4z^{7}a^{-3}+2z^{5}a^{-3}+a^{3}z+5za^{-3}-2a^{-3}z^{-1}+2z^{8}a^{-2}-10z^{6}a^{-2}+14z^{4}a^{-2}+a^{2}z^{2}-9z^{2}a^{-2}+3a^{-2}+2z^{7}a^{-1}-9z^{5}a^{-1}+2az^{3}+10z^{3}a^{-1}-4az-4za^{-1}+az^{-1}+z^{6}-3z^{4}+z^{2}-1$ (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   \ -2 -1 0 1 2 3 4 5 6 7 χ 14                   1 -1 12                     0 10               2 1   -1 8             1       1 6           1 2       1 4         2 1         1 2         1           1 0     2 2             0 -2     1               1 -4 1 1                 0 -6 1                   1

Integral Khovanov Homology (db, data source)

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

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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