L11n196

L11n195

L11n197

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n196 at Knotilus!

Link Presentations

[edit Notes on L11n196's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X13,21,14,20 X16,9,17,10 X10,19,11,20 X15,7,16,22 X21,15,22,14 X18,5,19,6 X2738 X4,11,5,12 X6,17,1,18
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 4, -5, 10, -2, -3, 7, -6, -4, 11, -8, 5, 3, -7, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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