L11n140

L11n139

L11n141

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n140 at Knotilus!

Link Presentations

[edit Notes on L11n140'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 X15,20,16,21 X13,22,14,7 X21,14,22,15 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 $u$, $v$, $w$, ...)	${\frac {u^{2}v^{2}-u^{2}v-3uv^{2}+5uv-3u-v+1}{uv}}$ (db)
Jones polynomial	${\frac {5}{q^{9/2}}}-{\frac {5}{q^{7/2}}}+{\frac {3}{q^{5/2}}}-{\frac {2}{q^{3/2}}}+{\frac {2}{q^{17/2}}}-{\frac {3}{q^{15/2}}}+{\frac {4}{q^{13/2}}}-{\frac {6}{q^{11/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{9}z^{-1}-z^{3}a^{7}+2a^{7}z^{-1}+z^{5}a^{5}+3z^{3}a^{5}+3za^{5}-2z^{3}a^{3}-4za^{3}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$3a^{10}z^{4}-7a^{10}z^{2}+2a^{10}+a^{9}z^{7}-a^{9}z^{5}-2a^{9}z^{3}+2a^{9}z-a^{9}z^{-1}+a^{8}z^{8}-3a^{8}z^{6}+9a^{8}z^{4}-13a^{8}z^{2}+5a^{8}+3a^{7}z^{7}-7a^{7}z^{5}+6a^{7}z^{3}+2a^{7}z-2a^{7}z^{-1}+a^{6}z^{8}-2a^{6}z^{6}+6a^{6}z^{4}-6a^{6}z^{2}+3a^{6}+2a^{5}z^{7}-6a^{5}z^{5}+11a^{5}z^{3}-5a^{5}z+a^{4}z^{6}-a^{4}+3a^{3}z^{3}-5a^{3}z+a^{3}z^{-1}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 χ -2               2 2 -4             2 1 -1 -6           3 1   2 -8         3 3     0 -10       3 2       1 -12     1 3         2 -14   2 3           -1 -16   1             1 -18 2               -2

Integral Khovanov Homology (db, data source)

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

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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