L11a509

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L11a508.gif

L11a508

L11a510.gif

L11a510

Contents

L11a509.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a509 at Knotilus!


Link Presentations

[edit Notes on L11a509's Link Presentations]

Planar diagram presentation X8192 X20,10,21,9 X14,5,15,6 X12,14,7,13 X16,8,17,7 X22,18,13,17 X10,4,11,3 X18,11,19,12 X6,15,1,16 X4,20,5,19 X2,21,3,22
Gauss code {1, -11, 7, -10, 3, -9}, {5, -1, 2, -7, 8, -4}, {4, -3, 9, -5, 6, -8, 10, -2, 11, -6}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation L11a509 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(3)-1) (-t(2) t(1)+t(2) t(3) t(1)-t(3) t(1)+t(1)+t(2)+t(3)-1) (-t(1) t(2)+t(1) t(3) t(2)-t(3) t(2)+t(2)-t(1) t(3)+t(3)-1)}{t(1) t(2) t(3)^{3/2}} (db)
Jones polynomial q^6-5 q^5+12 q^4-20 q^3+28 q^2-31 q+33-27 q^{-1} +21 q^{-2} -12 q^{-3} +5 q^{-4} - q^{-5} (db)
Signature 0 (db)
HOMFLY-PT polynomial z^8-a^2 z^6-2 z^6 a^{-2} +4 z^6-2 a^2 z^4-5 z^4 a^{-2} +z^4 a^{-4} +6 z^4-a^2 z^2-3 z^2 a^{-2} +z^2 a^{-4} +2 z^2+a^2+ a^{-2} -2+a^2 z^{-2} + a^{-2} z^{-2} -2 z^{-2} (db)
Kauffman polynomial z^6 a^{-6} -z^4 a^{-6} +5 z^7 a^{-5} +a^5 z^5-8 z^5 a^{-5} +3 z^3 a^{-5} +11 z^8 a^{-4} +5 a^4 z^6-22 z^6 a^{-4} -3 a^4 z^4+14 z^4 a^{-4} -4 z^2 a^{-4} +12 z^9 a^{-3} +12 a^3 z^7-18 z^7 a^{-3} -13 a^3 z^5+z^5 a^{-3} +3 a^3 z^3+4 z^3 a^{-3} +5 z^{10} a^{-2} +18 a^2 z^8+18 z^8 a^{-2} -28 a^2 z^6-62 z^6 a^{-2} +16 a^2 z^4+49 z^4 a^{-2} -3 a^2 z^2-11 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -2 a^2-2 a^{-2} +15 a z^9+27 z^9 a^{-1} -15 a z^7-50 z^7 a^{-1} -4 a z^5+19 z^5 a^{-1} +4 a z^3+2 z^3 a^{-1} +2 a z+2 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +5 z^{10}+25 z^8-72 z^6+53 z^4-10 z^2+2 z^{-2} -3 (db)

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r).   
\ r
  \  
j \
-5-4-3-2-10123456χ
13           11
11          4 -4
9         81 7
7        124  -8
5       168   8
3      1613    -3
1     1715     2
-1    1218      6
-3   915       -6
-5  413        9
-7 18         -7
-9 4          4
-111           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r=-1 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=0 {\mathbb Z}^{18}\oplus{\mathbb Z}_2^{15} {\mathbb Z}^{17}
r=1 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{16} {\mathbb Z}^{16}
r=2 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{15} {\mathbb Z}^{16}
r=3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=6 {\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).

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L11a508.gif

L11a508

L11a510.gif

L11a510