L11n73

L11n72

L11n74

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n73 at Knotilus!

Link Presentations

[edit Notes on L11n73's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X16,8,17,7 X17,22,18,5 X11,18,12,19 X21,12,22,13 X13,20,14,21 X19,14,20,15 X8,16,9,15 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, -5, 6, -7, 8, 9, -3, -4, 5, -8, 7, -6, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2uv^{3}-5uv^{2}+4uv-2u-2v^{3}+4v^{2}-5v+2}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	${\frac {9}{q^{9/2}}}-{\frac {9}{q^{7/2}}}+{\frac {7}{q^{5/2}}}-{\frac {6}{q^{3/2}}}+{\frac {1}{q^{17/2}}}-{\frac {3}{q^{15/2}}}+{\frac {5}{q^{13/2}}}-{\frac {8}{q^{11/2}}}-{\sqrt {q}}+{\frac {3}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}\left(-z^{3}\right)-a^{7}z+a^{5}z^{5}+2a^{5}z^{3}+2a^{5}z+a^{5}z^{-1}+a^{3}z^{5}+a^{3}z^{3}-2a^{3}z-a^{3}z^{-1}-az^{3}-az$ (db)
Kauffman polynomial	$a^{10}z^{4}-2a^{10}z^{2}+3a^{9}z^{5}-5a^{9}z^{3}+a^{9}z+a^{8}z^{8}-2a^{8}z^{6}+5a^{8}z^{4}-3a^{8}z^{2}+a^{7}z^{9}-2a^{7}z^{7}+5a^{7}z^{5}-2a^{7}z^{3}+a^{7}z+4a^{6}z^{8}-10a^{6}z^{6}+12a^{6}z^{4}-2a^{6}z^{2}+a^{5}z^{9}+2a^{5}z^{7}-7a^{5}z^{5}+8a^{5}z^{3}-4a^{5}z+a^{5}z^{-1}+3a^{4}z^{8}-5a^{4}z^{6}+2a^{4}z^{4}-a^{4}+4a^{3}z^{7}-8a^{3}z^{5}+3a^{3}z^{3}-3a^{3}z+a^{3}z^{-1}+3a^{2}z^{6}-6a^{2}z^{4}+a^{2}z^{2}+az^{5}-2az^{3}+az$ (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 1 2 χ 2                   1 1 0                 2   -2 -2               4 1   3 -4             4 3     -1 -6           5 3       2 -8         4 4         0 -10       4 5           -1 -12     2 5             3 -14   1 3               -2 -16   2                 2 -18 1                   -1

Integral Khovanov Homology (db, data source)

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

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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