L10n63

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L10n62

L10n64

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Image:L10n63.gif
(Knotscape image) 		See the full Thistlethwaite Link Table (up to 11 crossings).

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[edit] Link Presentations

[edit Notes on L10n63's Link Presentations]

Planar diagram presentation X12,1,13,2 X7,17,8,16 X3948 X17,2,18,3 X5,14,6,15 X11,6,12,7 X9,18,10,19 X15,11,16,20 X10,13,1,14 X19,5,20,4
Gauss code {1, 4, -3, 10, -5, 6, -2, 3, -7, -9}, {-6, -1, 9, 5, -8, 2, -4, 7, -10, 8}
A Braid Representative
Image:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart4.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart4.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gif
Image:BraidPart0.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gif
Image:BraidPart0.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gif
A Morse Link Presentation Image:L10n63_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u v+1) \left(u^2 v^2-2 u^2 v+u^2-2 u v^2+3 u v-2 u+v^2-2 v+1\right)}{u^{3/2} v^{3/2}} (db)
Jones polynomial -q^{7/2}+4 q^{5/2}-7 q^{3/2}+9 \sqrt{q}-\frac{11}{\sqrt{q}}+\frac{10}{q^{3/2}}-\frac{9}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{3}{q^{9/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a5z + a5z−1−a3z5−3a3z3−3a3z−a3z−1 + az7 + 4az5−z5a−1 + 4az3−2z3a−1 (db)
Kauffman polynomial −3a2z8−3z8−6a3z7−12az7−6z7a−1−3a4z6−a2z6−4z6a−2−2z6 + 13a3z5 + 26az5 + 12z5a−1−z5a−3 + 6a2z4 + 7z4a−2 + 13z4−6a5z3−18a3z3−18az3−5z3a−1 + z3a−3−3a2z2−2z2a−2−5z2 + 6a5z + 9a3z + 4az + za−1 + a4−a5z−1−a3z−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 L10n63. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.    <table border=1> <tr align=center> <td width=15.3846%><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=7.69231%>-4</td><td width=7.69231%>-3</td><td width=7.69231%>-2</td><td width=7.69231%>-1</td><td width=7.69231%>0</td><td width=7.69231%>1</td><td width=7.69231%>2</td><td width=7.69231%>3</td><td width=7.69231%>4</td><td width=15.3846%>χ</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 bgcolor=yellow>1</td><td>1</td></tr> <tr align=center><td>6</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>4</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>4</td><td bgcolor=yellow>1</td><td> </td><td>3</td></tr> <tr align=center><td>2</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>5</td><td bgcolor=yellow>3</td><td> </td><td> </td><td>-2</td></tr> <tr align=center><td>0</td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>6</td><td bgcolor=yellow>4</td><td> </td><td> </td><td> </td><td>2</td></tr> <tr align=center><td>-2</td><td> </td><td> </td><td> </td><td bgcolor=yellow>5</td><td bgcolor=yellow>6</td><td> </td><td> </td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>-4</td><td> </td><td> </td><td bgcolor=yellow>4</td><td bgcolor=yellow>5</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-1</td></tr> <tr align=center><td>-6</td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow>5</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>3</td></tr> <tr align=center><td>-8</td><td bgcolor=yellow>1</td><td bgcolor=yellow>4</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>-3</td></tr> <tr align=center><td>-10</td><td bgcolor=yellow>3</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>3</td></tr> </table>
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i = −2 i = 0
r = −4 {\mathbb Z}^{3} {\mathbb Z}
r = −3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = −2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = −1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r = 0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r = 1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r = 2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r = 3 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r = 4 {\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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L10n62

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