L7a1

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L6n1

L7a2

Contents

Image:L7a1.gif
(Knotscape image) 		See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L7a1's page at Knotilus.

Visit L7a1's page at the original Knot Atlas.

[edit L7a1 Quick Notes]

L7a1 is 7^2_6 in the Rolfsen table of links.

[edit L7a1 Further Notes and Views]


[edit] Link Presentations

[edit Notes on L7a1's Link Presentations]

Planar diagram presentation X6172 X12,7,13,8 X4,13,1,14 X10,6,11,5 X8493 X14,10,5,9 X2,12,3,11
Gauss code {1, -7, 5, -3}, {4, -1, 2, -5, 6, -4, 7, -2, 3, -6}
A Braid Representative
Image:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart3.gifImage:BraidPart0.gifImage:BraidPart0.gif
Image:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart4.gifImage:BraidPart1.gifImage:BraidPart1.gif
Image:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart0.gifImage:BraidPart2.gifImage:BraidPart2.gif
A Morse Link Presentation Image:L7a1_ML.gif

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1) \left(v^2-v+1\right)}{\sqrt{u} v^{3/2}} (db)
Jones polynomial -q^{9/2}+3 q^{7/2}-4 q^{5/2}+\frac{1}{q^{5/2}}+4 q^{3/2}-\frac{3}{q^{3/2}}-5 \sqrt{q}+\frac{3}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial −z3a−3−za−3 + z5a−1−az3 + 3z3a−1−az + 2za−1 + az−1−a−1z−1 (db)
Kauffman polynomial z3a−5 + 3z4a−4−2z2a−4 + 4z5a−3−5z3a−3 + 2za−3 + 2z6a−2 + a2z4 + z4a−2−a2z2−3z2a−2 + 3az5 + 7z5a−1−6az3−12z3a−1 + 2az + 4za−1 + az−1 + a−1z−1 + 2z6−z4−2z2−1 (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 L7a1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.    <table border=1> <tr align=center> <td width=16.6667%><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=8.33333%>-3</td><td width=8.33333%>-2</td><td width=8.33333%>-1</td><td width=8.33333%>0</td><td width=8.33333%>1</td><td width=8.33333%>2</td><td width=8.33333%>3</td><td width=8.33333%>4</td><td width=16.6667%>χ</td></tr> <tr align=center><td>10</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 bgcolor=yellow>2</td><td bgcolor=yellow> </td><td>-2</td></tr> <tr align=center><td>6</td><td> </td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow>1</td><td> </td><td>1</td></tr> <tr align=center><td>4</td><td> </td><td> </td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow>2</td><td> </td><td> </td><td>0</td></tr> <tr align=center><td>2</td><td> </td><td> </td><td> </td><td bgcolor=yellow>3</td><td bgcolor=yellow>2</td><td> </td><td> </td><td> </td><td>1</td></tr> <tr align=center><td>0</td><td> </td><td> </td><td bgcolor=yellow>2</td><td bgcolor=yellow>4</td><td> </td><td> </td><td> </td><td> </td><td>2</td></tr> <tr align=center><td>-2</td><td> </td><td bgcolor=yellow>1</td><td bgcolor=yellow>1</td><td> </td><td> </td><td> </td><td> </td><td> </td><td>0</td></tr> <tr align=center><td>-4</td><td bgcolor=yellow> </td><td bgcolor=yellow>2</td><td> </td><td> </td><td> </td><td> </td><td> </td><td> </td><td>2</td></tr> <tr align=center><td>-6</td><td bgcolor=yellow>1</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 = 0 i = 2
r = −3 {\mathbb Z}
r = −2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r = −1 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = 0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}^{3}
r = 1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = 2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r = 3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

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L6n1

L7a2

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