L11n142

L11n141

L11n143

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n142 at Knotilus!

Link Presentations

[edit Notes on L11n142's Link Presentations]

Planar diagram presentation	X8192 X18,11,19,12 X3,10,4,11 X17,3,18,2 X12,5,13,6 X6718 X9,16,10,17 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 $u$, $v$, $w$, ...)	$-{\frac {2u^{2}v^{2}-2u^{2}v-3uv^{2}+7uv-3u-2v+2}{uv}}$ (db)
Jones polynomial	$q^{5/2}-3q^{3/2}+4{\sqrt {q}}-{\frac {7}{\sqrt {q}}}+{\frac {7}{q^{3/2}}}-{\frac {7}{q^{5/2}}}+{\frac {6}{q^{7/2}}}-{\frac {4}{q^{9/2}}}+{\frac {2}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^{3}a^{5}+2za^{5}+a^{5}z^{-1}-z^{5}a^{3}-3z^{3}a^{3}-4za^{3}-2a^{3}z^{-1}-z^{5}a-2z^{3}a+2az^{-1}+z^{3}a^{-1}+za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-3a^{7}z^{3}+a^{7}z+2a^{6}z^{6}-5a^{6}z^{4}+a^{6}z^{2}+3a^{5}z^{7}-9a^{5}z^{5}+8a^{5}z^{3}-4a^{5}z+a^{5}z^{-1}+3a^{4}z^{8}-11a^{4}z^{6}+16a^{4}z^{4}-8a^{4}z^{2}+a^{3}z^{9}-8a^{3}z^{5}+18a^{3}z^{3}-10a^{3}z+2a^{3}z^{-1}+4a^{2}z^{8}-16a^{2}z^{6}+27a^{2}z^{4}+z^{4}a^{-2}-13a^{2}z^{2}-2z^{2}a^{-2}+a^{2}+az^{9}-3az^{7}+5az^{5}+3z^{5}a^{-1}+az^{3}-6z^{3}a^{-1}-4az+za^{-1}+2az^{-1}+a^{-1}z^{-1}+z^{8}-3z^{6}+7z^{4}-6z^{2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 6                   1 -1 4                 2   2 2               2 1   -1 0             5 2     3 -2           3 3       0 -4         4 4         0 -6       3 4           1 -8     1 3             -2 -10   1 3               2 -12   1                 -1 -14 1                   1

Integral Khovanov Homology (db, data source)

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

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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