L11n442

L11n441

L11n443

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n442 at Knotilus!

Link Presentations

[edit Notes on L11n442's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uvw^{2}x-uvw^{2}+uvwx^{2}-4uvwx+2uvw-uvx^{2}+2uvx-uv-uwx^{2}+2uwx+ux^{2}-ux-vw^{2}x+vw^{2}+2vwx-vw+w^{2}\left(-x^{2}\right)+2w^{2}x-w^{2}+2wx^{2}-4wx+w-x^{2}+x}{{\sqrt {u}}{\sqrt {v}}wx}}$ (db)
Jones polynomial	$-10q^{9/2}+10q^{7/2}-14q^{5/2}+10q^{3/2}-{\frac {3}{q^{3/2}}}+q^{15/2}-3q^{13/2}+6q^{11/2}-10{\sqrt {q}}+{\frac {5}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$2z^{5}a^{-3}-5z^{3}a^{-1}+7z^{3}a^{-3}-3z^{3}a^{-5}+3az-11za^{-1}+13za^{-3}-6za^{-5}+za^{-7}+3az^{-1}-9a^{-1}z^{-1}+10a^{-3}z^{-1}-5a^{-5}z^{-1}+a^{-7}z^{-1}+az^{-3}-3a^{-1}z^{-3}+3a^{-3}z^{-3}-a^{-5}z^{-3}$ (db)
Kauffman polynomial	$z^{6}a^{-8}-3z^{4}a^{-8}+3z^{2}a^{-8}-a^{-8}+3z^{7}a^{-7}-9z^{5}a^{-7}+9z^{3}a^{-7}-6za^{-7}+2a^{-7}z^{-1}+3z^{8}a^{-6}-3z^{6}a^{-6}-11z^{4}a^{-6}+14z^{2}a^{-6}-6a^{-6}+z^{9}a^{-5}+10z^{7}a^{-5}-40z^{5}a^{-5}+47z^{3}a^{-5}-a^{-5}z^{-3}-30za^{-5}+11a^{-5}z^{-1}+8z^{8}a^{-4}-14z^{6}a^{-4}-9z^{4}a^{-4}+28z^{2}a^{-4}+3a^{-4}z^{-2}-18a^{-4}+z^{9}a^{-3}+14z^{7}a^{-3}-53z^{5}a^{-3}+74z^{3}a^{-3}-3a^{-3}z^{-3}-49za^{-3}+18a^{-3}z^{-1}+5z^{8}a^{-2}-7z^{6}a^{-2}-4z^{4}a^{-2}+24z^{2}a^{-2}+6a^{-2}z^{-2}-21a^{-2}+7z^{7}a^{-1}-22z^{5}a^{-1}+6az^{3}+42z^{3}a^{-1}-az^{-3}-3a^{-1}z^{-3}-10az-35za^{-1}+5az^{-1}+14a^{-1}z^{-1}+3z^{6}-3z^{4}+7z^{2}+3z^{-2}-9$ (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   \ -2 -1 0 1 2 3 4 5 6 7 χ 16                   1 -1 14                 2   2 12               4 1   -3 10             6 2     4 8           5 5       0 6         9 5         4 4       4 8           4 2     6 6             0 0   2 7               5 -2 1 3                 -2 -4 3                   3

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-2$ ${\mathbb {Z} }^{3}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{6}$ $r=1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{9}$ $r=3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{6}$ $r=5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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