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---
layout: default
---
<h2>Classifying, realizing ...</h2>
<h3>... all combinatorial 3-spheres and 4-polytopes with up to 9 vertices</h3>
All combinatorial types are preceded by the flag f-vector <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><msub><mi>f</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>f</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>f</mi><mn>2</mn></msub><mo separator="true">,</mo><msub><mi>f</mi><mn>3</mn></msub><mo>)</mo><mo separator="true">,</mo><msub><mi>f</mi><mrow><mn>0</mn><mo separator="true">,</mo><mn>3</mn></mrow></msub></mrow><annotation encoding="application/x-tex">(f_0, f_1, f_2, f_3 ), f_{0,3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1.036108em;vertical-align:-0.286108em;"></span><span class="base"><span class="mopen">(</span><span class="mord"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mpunct">,</span><span class="mord rule" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mpunct">,</span><span class="mord rule" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mpunct">,</span><span class="mord rule" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mord rule" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span><span class="mpunct mtight">,</span><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"></span></span></span></span></span></span></span></span>.
<table class="padded-table"><tr><td></td><td>n=5</td><td>n=6</td><td>n=7</td><td>n=8</td><td>n=9</td><td>5<=n<=9</td></tr>
<tr><td>combinatorial 3-spheres
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/classification/all3spheres5vertices.txt">1</a></td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/classification/all3spheres6vertices.txt">4</a></td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/classification/all3spheres7vertices.txt">31</a></td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/classification/all3spheres8vertices.txt">1336</a></td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/classification/all3spheres9vertices.txt">316014</a></td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/classification/all3spheres.tar.gz">tar.gz</a></td>
</tr>
<tr><td>4-polytopes</td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/classification/all4polytopes5vertices.txt">1</a></td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/classification/all4polytopes6vertices.txt">4</a></td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/classification/all4polytopes7vertices.txt">31</a></td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/classification/all4polytopes8vertices.txt">1294</a></td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/classification/all4polytopes9vertices.txt">274148</a></td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/classification/all4polytopes.tar.gz">tar.gz</a></td>
</tr>
</table>
<center><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/classification/combi3spheres9verts.tar.gz">all combinatorial 3-spheres, 9 vertices, with rational realization or non-realizability certificates</a></center>
For n=9 vertices, for each combinatorial sphere, we provide either a realization or a certificate for non-realizability.
<br>
Each line contains a combinatorial sphere.
<br>
<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><msub><mi>f</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>f</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>f</mi><mn>2</mn></msub><mo separator="true">,</mo><msub><mi>f</mi><mn>3</mn></msub><mo>)</mo><mo separator="true">,</mo><msub><mi>f</mi><mrow><mn>0</mn><mo separator="true">,</mo><mn>3</mn></mrow></msub></mrow><annotation encoding="application/x-tex">(f_0, f_1, f_2, f_3 ), f_{0,3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1.036108em;vertical-align:-0.286108em;"></span><span class="base"><span class="mopen">(</span><span class="mord"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mpunct">,</span><span class="mord rule" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mpunct">,</span><span class="mord rule" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mpunct">,</span><span class="mord rule" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mord rule" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathit" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span><span class="mpunct mtight">,</span><span class="mord mtight">3</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"></span></span></span></span></span></span></span></span>, [polytope|nonrealizable] type, additional data.
<br>
If the sphere is the boundary of a polytope, we provide rational coordinates of such a polytope.
<br>
Otherwise we provide a non-realizability certificate of a certain type.
where "type" can be one of three values:
<ol>
<li> this combinatorial type has a partial chirotope that contradicts Graßmann-Plücker: the partial chirotope and the GP-relation violated follows
</li>
<li> this combinatorial type has a partial chirotope which can't be completed consistently. Relevant GP-relations and the contradiction follow
</li>
<li> the completed chirotope admits a biquadratic final polynomial. We provide the completed chirotope and the linear program, which is infeasible
</li>
</ol>
<hr>
<h3>... and inscribing simplicial 3-spheres with up to 10 vertices</h3>
Numbering corresponds to the <a href="http://page.math.tu-berlin.de/~lutz/stellar/3-manifolds.html#n_vertices">enumeration of Frank Lutz</a>.
This is a table with the number simplicial 3-spheres with n vertices. Click on the file to download a file with rational realizations. Most of the realizations are inscribed in the unit sphere, the second row states how many of the realizations are inscribed.
<table class="padded-table"><tr><td></td><td>n=5</td><td>n=6</td><td>n=7</td><td>n=8</td><td>n=9</td><td>n=10</td></td></tr>
<tr><td>d=4</td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/inscribe/polytopalsimplicialspheres5_4.txt">1</a></td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/inscribe/polytopalsimplicialspheres6_4.txt">2</td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/inscribe/polytopalsimplicialspheres7_4.txt">5</a></td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/inscribe/polytopalsimplicialspheres8_4.txt">37</a></td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/inscribe/polytopalsimplicialspheres9_4.txt">1142</a></td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/inscribe/polytopalsimplicialspheres10_4.txt">162004</a></td>
</tr>
<tr><td>inscribed</td><td>1</td><td>2</td><td>5</td> <td>37</td> <td>1140</td> <td>161978</td>
</tr>
</table>
<hr>
<h3>... neighborly uniform oriented matroids</h3>
Numbering corresponds to the <a href="https://sites.google.com/site/hmiyata1984/neighborly_polytopes">enumeration by Hiroyuki Miyata (宮田 洋行)</a>.
Here is a table with the number of (simplicial) neighborly d-polytopes with n vertices for small d and n. Click on the number to download a file with rational realizations of these polytopes on the sphere. Some of the files are rather large.
<table class="padded-table"><tr><td></td><td>n=5</td><td>n=6</td><td>n=7</td><td>n=8</td><td>n=9</td><td>n=10</td><td>n=11</td><td>n=12</td></tr>
<tr><td>d=4</td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/inscribe/neighborly/neighborly5_4inscribed.txt">1</a></td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/inscribe/neighborly/neighborly6_4inscribed.txt">1</a></td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/inscribe/neighborly/neighborly7_4inscribed.txt">1</a></td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/inscribe/neighborly/neighborly8_4inscribed.txt">3</a></td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/inscribe/neighborly/neighborly9_4inscribed.txt">23</a></td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/inscribe/neighborly/neighborly10_4inscribed.txt">431</a></td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/inscribe/neighborly/neighborly11_4inscribed.txt">13935</a></td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/inscribe/neighborly/neighborly12_4inscribed.txt">556061</a></td>
</tr>
<tr><td>d=5</td>
<td></td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/inscribe/neighborly/neighborly6_5inscribed.txt">1</a></td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/inscribe/neighborly/neighborly7_5inscribed.txt">1</a></td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/inscribe/neighborly/neighborly8_5inscribed.txt">2</a></td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/inscribe/neighborly/neighborly9_5inscribed.txt">126</a></td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/inscribe/neighborly/neighborly10_5inscribed.txt">159375</a></td>
<td></td>
<td></td>
</tr>
<tr><td>d=6</td>
<td></td>
<td></td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/inscribe/neighborly/neighborly7_6inscribed.txt">1</a></td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/inscribe/neighborly/neighborly8_6inscribed.txt">1</a></td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/inscribe/neighborly/neighborly9_6inscribed.txt">1</a></td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/inscribe/neighborly/neighborly10_6inscribed.txt">37</a></td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/inscribe/neighborly/neighborly11_6inscribed.txt">42099</a></td>
<td></td>
</tr>
<tr><td>d=7</td>
<td></td>
<td></td>
<td></td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/inscribe/neighborly/neighborly8_7inscribed.txt">1</a></td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/inscribe/neighborly/neighborly9_7inscribed.txt">1</a></td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/inscribe/neighborly/neighborly10_7inscribed.txt">4</a></td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/inscribe/neighborly/neighborly11_7inscribed.txt">35993</a></td>
<td></td>
</tr>
<!--<td>d=8</td>
<td></td>
<td></td>
<td></td>
<td></td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/inscribe/neighborly/neighborly9_8inscribed.txt">1</a></td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/inscribe/neighborly/neighborly10_8inscribed.txt">1</a></td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/inscribe/neighborly/neighborly11_8inscribed.txt">1</a></td>
<td><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/inscribe/neighborly/neighborly12_8inscribed.txt">2592</a></td>
</tr>-->
</table>
2-neighborly simplicial 6-polytopes with 10 vertices: <a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/inscribe/neighborly/2n10_6inscribed.txt">4523</a>
<hr>
<h3>... simplicial 3-spheres with small valence</h3>
Numbering corresponds to the <a href="http://page.math.tu-berlin.de/~lutz/stellar/3-manifolds.html#small_valence">enumeration of Frank Lutz</a>.
We provide rational realizations on the unit sphere.
<center><a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/inscribe/smallvalencerealizations.txt">4759 inscriptions of simplicial 3-spheres with small valence</a></center>
<!--<hr><center>all the files from this site as compressed archive: <a href="ftp://ftp.imp.fu-berlin.de/pub/moritz/inscribe/realizations.tar.gz">realizations.tar.gz</a> (~300MB)</center>-->