L11n160

L11n159

L11n161

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n160 at Knotilus!

Link Presentations

[edit Notes on L11n160's Link Presentations]

Planar diagram presentation	X8192 X16,7,17,8 X10,4,11,3 X2,15,3,16 X14,10,15,9 X11,19,12,18 X5,13,6,12 X6,21,1,22 X20,14,21,13 X22,17,7,18 X19,4,20,5
Gauss code	{1, -4, 3, 11, -7, -8}, {2, -1, 5, -3, -6, 7, 9, -5, 4, -2, 10, 6, -11, -9, 8, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(uv-u-v+2)(2uv-u-v+1)}{uv}}$ (db)
Jones polynomial	$q^{9/2}-3q^{7/2}+5q^{5/2}-7q^{3/2}+8{\sqrt {q}}-{\frac {9}{\sqrt {q}}}+{\frac {7}{q^{3/2}}}-{\frac {6}{q^{5/2}}}+{\frac {3}{q^{7/2}}}-{\frac {1}{q^{9/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{3}+z^{3}a^{-3}+a^{3}z+za^{-3}+a^{3}z^{-1}-az^{5}-z^{5}a^{-1}-2az^{3}-2z^{3}a^{-1}-2az-za^{-1}-az^{-1}$ (db)
Kauffman polynomial	$-2az^{9}-2z^{9}a^{-1}-3a^{2}z^{8}-4z^{8}a^{-2}-7z^{8}-a^{3}z^{7}+5az^{7}+3z^{7}a^{-1}-3z^{7}a^{-3}+11a^{2}z^{6}+14z^{6}a^{-2}-z^{6}a^{-4}+26z^{6}-3az^{5}+7z^{5}a^{-1}+10z^{5}a^{-3}-3a^{4}z^{4}-18a^{2}z^{4}-12z^{4}a^{-2}+3z^{4}a^{-4}-30z^{4}-a^{5}z^{3}-a^{3}z^{3}-3az^{3}-10z^{3}a^{-1}-7z^{3}a^{-3}+3a^{4}z^{2}+8a^{2}z^{2}+4z^{2}a^{-2}-z^{2}a^{-4}+10z^{2}+a^{5}z+3za^{-1}+2za^{-3}-a^{2}+a^{3}z^{-1}+az^{-1}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 10                   1 -1 8                 2   2 6               3 1   -2 4             4 2     2 2           4 3       -1 0         5 4         1 -2       3 5           2 -4     3 4             -1 -6   1 4               3 -8   2                 -2 -10 1                   1

Integral Khovanov Homology (db, data source)

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

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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