L11a177

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L11a176

L11a178

Contents

Image:L11a177.gif
(Knotscape image) 		See the full Thistlethwaite Link Table (up to 11 crossings).

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[edit L11a177 Quick Notes]
[edit L11a177 Further Notes and Views]


[edit] Link Presentations

[edit Notes on L11a177's Link Presentations]

Planar diagram presentation X8192 X10,4,11,3 X16,7,17,8 X18,10,19,9 X20,11,21,12 X22,15,7,16 X12,21,13,22 X14,6,15,5 X4,14,5,13 X6,18,1,17 X2,19,3,20
Gauss code {1, -11, 2, -9, 8, -10}, {3, -1, 4, -2, 5, -7, 9, -8, 6, -3, 10, -4, 11, -5, 7, -6}
A Braid Representative
Image:BraidPart1.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart1.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart2.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart2.gifImage:BraidPart3.gifImage:BraidPart3.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart4.gif
A Morse Link Presentation Image:L11a177_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{\left(v^2-v+1\right) \left(u v^2-u v+u+v-1\right) \left(u v^2-u v-v^2+v-1\right)}{u v^3} (db)
Jones polynomial q^{9/2}-\frac{9}{q^{9/2}}-4 q^{7/2}+\frac{16}{q^{7/2}}+9 q^{5/2}-\frac{21}{q^{5/2}}-16 q^{3/2}+\frac{24}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{4}{q^{11/2}}+20 \sqrt{q}-\frac{25}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a3z7 + 4a3z5 + 5a3z3 + a3z−a3z−1−az9−6az7 + z7a−1−13az5 + 4z5a−1−11az3 + 5z3a−1−az + za−1 + 3az−1−2a−1z−1 (db)
Kauffman polynomial −3a2z10−3z10−8a3z9−15az9−7z9a−1−10a4z8−11a2z8−7z8a−2−8z8−8a5z7 + 6a3z7 + 28az7 + 10z7a−1−4z7a−3−4a6z6 + 13a4z6 + 26a2z6 + 14z6a−2−z6a−4 + 24z6−a7z5 + 11a5z5 + a3z5−20az5 + 9z5a−3 + 5a6z4−5a4z4−14a2z4−6z4a−2 + 2z4a−4−12z4 + a7z3−5a5z3 + 2a3z3 + 12az3−2z3a−1−6z3a−3−2a6z2−a4z2−z2a−2−z2a−4−z2−2a3z−az + 2za−1 + za−3 + a4 + 3a2 + 3−a3z−1−3az−1−2a−1z−1 (db)

[edit] Khovanov Homology

The coefficients of the monomials trqj are shown, along with their alternating sums χ (fixed j, alternation over r). The squares with yellow highlighting are those on the "critical diagonals", where j−2r = s + 1 or j−2r = s−1, where s = -1 is the signature of L11a177. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.    <table border=1> <tr align=center> <td width=12.5%><table cellpadding=0 cellspacing=0> <tr><td>\</td><td> </td><td>r</td></tr> <tr><td> </td><td> \ </td><td> </td></tr> <tr><td>j</td><td> </td><td>\</td></tr> </table></td> <td width=6.25%>-6</td><td width=6.25%>-5</td><td width=6.25%>-4</td><td width=6.25%>-3</td><td width=6.25%>-2</td><td width=6.25%>-1</td><td width=6.25%>0</td><td width=6.25%>1</td><td width=6.25%>2</td><td width=6.25%>3</td><td width=6.25%>4</td><td width=6.25%>5</td><td width=12.5%>χ</td></tr> <tr align=center><td>10</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>1</td><td>-1</td></tr> <tr align=center><td>8</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>3</td><td bgcolor=yellow> </td><td>3</td></tr> <tr align=center><td>6</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>6</td><td bgcolor=yellow>1</td><td> </td><td>-5</td></tr> <tr align=center><td>4</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>10</td><td bgcolor=yellow>3</td><td> </td><td> </td><td>7</td></tr> <tr align=center><td>2</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>11</td><td bgcolor=yellow>7</td><td> </td><td> </td><td> </td><td>-4</td></tr> <tr align=center><td>0</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>14</td><td bgcolor=yellow>9</td><td> </td><td> </td><td> </td><td> </td><td>5</td></tr> <tr align=center><td>-2</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>11</td><td bgcolor=yellow>12</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>-4</td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>10</td><td bgcolor=yellow>13</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-3</td></tr> <tr align=center><td>-6</td><td> </td><td> </td><td> </td><td bgcolor=yellow>6</td><td bgcolor=yellow>11</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>5</td></tr> <tr align=center><td>-8</td><td> </td><td> </td><td bgcolor=yellow>3</td><td bgcolor=yellow>10</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-7</td></tr> <tr align=center><td>-10</td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>6</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>5</td></tr> <tr align=center><td>-12</td><td bgcolor=yellow> </td><td bgcolor=yellow>3</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-3</td></tr> <tr align=center><td>-14</td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> </table>
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = −2 i = 0
r = −6 {\mathbb Z}
r = −5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = −3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = −2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r = −1 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r = 0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{14}
r = 1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r = 2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r = 3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r = 4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = 5 {\mathbb Z}_2 {\mathbb Z}

[edit] Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

[edit] Modifying This Page

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L11a176

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