L11n182

L11n181

L11n183

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n182 at Knotilus!

Link Presentations

[edit Notes on L11n182's Link Presentations]

Planar diagram presentation	X8192 X9,21,10,20 X14,5,15,6 X16,8,17,7 X3,10,4,11 X22,14,7,13 X18,12,19,11 X12,18,13,17 X6,15,1,16 X4,21,5,22 X19,2,20,3
Gauss code	{1, 11, -5, -10, 3, -9}, {4, -1, -2, 5, 7, -8, 6, -3, 9, -4, 8, -7, -11, 2, 10, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{2}v^{3}-3u^{2}v^{2}+3u^{2}v+2uv^{4}-5uv^{3}+7uv^{2}-5uv+2u+3v^{3}-3v^{2}+v}{uv^{2}}}$ (db)
Jones polynomial	$2q^{5/2}-5q^{3/2}+8{\sqrt {q}}-{\frac {11}{\sqrt {q}}}+{\frac {12}{q^{3/2}}}-{\frac {12}{q^{5/2}}}+{\frac {9}{q^{7/2}}}-{\frac {7}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+a^{5}z+a^{5}z^{-1}-a^{3}z^{5}-a^{3}z^{3}-2az^{5}-6az^{3}+2z^{3}a^{-1}-7az-2az^{-1}+3za^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-2a^{7}z^{3}+3a^{6}z^{6}-5a^{6}z^{4}+6a^{5}z^{7}-16a^{5}z^{5}+16a^{5}z^{3}-8a^{5}z+a^{5}z^{-1}+5a^{4}z^{8}-10a^{4}z^{6}+6a^{4}z^{4}+a^{4}z^{2}-a^{4}+2a^{3}z^{9}+2a^{3}z^{7}-10a^{3}z^{5}+10a^{3}z^{3}-2a^{3}z+8a^{2}z^{8}-21a^{2}z^{6}+27a^{2}z^{4}+3z^{4}a^{-2}-14a^{2}z^{2}-5z^{2}a^{-2}+3a^{2}+2a^{-2}+2az^{9}-3az^{7}+z^{7}a^{-1}+10az^{5}+3z^{5}a^{-1}-15az^{3}-7z^{3}a^{-1}+9az+3za^{-1}-2az^{-1}-a^{-1}z^{-1}+3z^{8}-8z^{6}+19z^{4}-20z^{2}+5$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 6                   2 -2 4                 3   3 2               5 2   -3 0             6 3     3 -2           7 6       -1 -4         5 5         0 -6       4 7           3 -8     3 5             -2 -10     4               4 -12 1 3                 -2 -14 1                   1

Integral Khovanov Homology (db, data source)

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

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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