L11n105

L11n104

L11n106

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n105 at Knotilus!

Link Presentations

[edit Notes on L11n105's Link Presentations]

Planar diagram presentation	X6172 X3,13,4,12 X7,16,8,17 X17,22,18,5 X13,18,14,19 X9,21,10,20 X19,14,20,15 X21,9,22,8 X15,10,16,11 X2536 X11,1,12,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 8, -6, 9, -11, 2, -5, 7, -9, 3, -4, 5, -7, 6, -8, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)^{3}\left(t(2)^{2}+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	${\frac {8}{q^{9/2}}}-{\frac {10}{q^{7/2}}}-q^{5/2}+{\frac {10}{q^{5/2}}}+3q^{3/2}-{\frac {11}{q^{3/2}}}+{\frac {2}{q^{13/2}}}-{\frac {5}{q^{11/2}}}-6{\sqrt {q}}+{\frac {8}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-za^{7}-a^{7}z^{-1}+z^{5}a^{5}+4z^{3}a^{5}+6za^{5}+3a^{5}z^{-1}-z^{7}a^{3}-5z^{5}a^{3}-10z^{3}a^{3}-10za^{3}-3a^{3}z^{-1}+2z^{5}a+7z^{3}a+7za+2az^{-1}-z^{3}a^{-1}-2za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$3a^{8}z^{2}-a^{8}+a^{7}z^{5}+5a^{7}z^{3}-3a^{7}z+a^{7}z^{-1}+5a^{6}z^{6}-7a^{6}z^{4}+6a^{6}z^{2}-2a^{6}+9a^{5}z^{7}-26a^{5}z^{5}+28a^{5}z^{3}-14a^{5}z+3a^{5}z^{-1}+7a^{4}z^{8}-18a^{4}z^{6}+11a^{4}z^{4}-3a^{4}z^{2}+2a^{3}z^{9}+7a^{3}z^{7}-42a^{3}z^{5}+48a^{3}z^{3}-22a^{3}z+3a^{3}z^{-1}+10a^{2}z^{8}-35a^{2}z^{6}+32a^{2}z^{4}-10a^{2}z^{2}+2a^{2}+2az^{9}-az^{7}+z^{7}a^{-1}-19az^{5}-4z^{5}a^{-1}+31az^{3}+6z^{3}a^{-1}-15az-4za^{-1}+2az^{-1}+a^{-1}z^{-1}+3z^{8}-12z^{6}+14z^{4}-4z^{2}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 6                   1 1 4                 2   -2 2               4 1   3 0             4 2     -2 -2           7 4       3 -4         5 6         1 -6       5 5           0 -8     3 5             2 -10   2 5               -3 -12   3                 3 -14 2                   -2

Integral Khovanov Homology (db, data source)

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

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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