L11a223

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L11a222.gif

L11a222

L11a224.gif

L11a224

Contents

L11a223.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a223 at Knotilus!


Link Presentations

[edit Notes on L11a223's Link Presentations]

Planar diagram presentation X8192 X14,9,15,10 X4758 X16,6,17,5 X18,16,19,15 X6,18,1,17 X22,19,7,20 X20,12,21,11 X10,22,11,21 X2,14,3,13 X12,4,13,3
Gauss code {1, -10, 11, -3, 4, -6}, {3, -1, 2, -9, 8, -11, 10, -2, 5, -4, 6, -5, 7, -8, 9, -7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a223 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1)^2 t(2)^4-2 t(1) t(2)^4+t(2)^4-4 t(1)^2 t(2)^3+10 t(1) t(2)^3-5 t(2)^3+7 t(1)^2 t(2)^2-17 t(1) t(2)^2+7 t(2)^2-5 t(1)^2 t(2)+10 t(1) t(2)-4 t(2)+t(1)^2-2 t(1)+1}{t(1) t(2)^2} (db)
Jones polynomial -16 q^{9/2}+21 q^{7/2}-\frac{1}{q^{7/2}}-25 q^{5/2}+\frac{4}{q^{5/2}}+25 q^{3/2}-\frac{10}{q^{3/2}}+q^{15/2}-4 q^{13/2}+9 q^{11/2}-22 \sqrt{q}+\frac{16}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^7 a^{-1} +a z^5-4 z^5 a^{-1} +3 z^5 a^{-3} +2 a z^3-9 z^3 a^{-1} +7 z^3 a^{-3} -3 z^3 a^{-5} +3 a z-7 z a^{-1} +7 z a^{-3} -3 z a^{-5} +z a^{-7} +a z^{-1} -2 a^{-1} z^{-1} +2 a^{-3} z^{-1} - a^{-5} z^{-1} (db)
Kauffman polynomial -2 z^{10} a^{-2} -2 z^{10} a^{-4} -7 z^9 a^{-1} -13 z^9 a^{-3} -6 z^9 a^{-5} -20 z^8 a^{-2} -16 z^8 a^{-4} -7 z^8 a^{-6} -11 z^8-9 a z^7-6 z^7 a^{-1} +9 z^7 a^{-3} +2 z^7 a^{-5} -4 z^7 a^{-7} -4 a^2 z^6+45 z^6 a^{-2} +41 z^6 a^{-4} +14 z^6 a^{-6} -z^6 a^{-8} +15 z^6-a^3 z^5+14 a z^5+33 z^5 a^{-1} +27 z^5 a^{-3} +18 z^5 a^{-5} +9 z^5 a^{-7} +4 a^2 z^4-28 z^4 a^{-2} -26 z^4 a^{-4} -8 z^4 a^{-6} +2 z^4 a^{-8} -8 z^4+a^3 z^3-11 a z^3-32 z^3 a^{-1} -32 z^3 a^{-3} -19 z^3 a^{-5} -7 z^3 a^{-7} -a^2 z^2+6 z^2 a^{-2} +5 z^2 a^{-4} +z^2 a^{-6} -z^2 a^{-8} +2 z^2+5 a z+13 z a^{-1} +13 z a^{-3} +7 z a^{-5} +2 z a^{-7} - a^{-2} -a z^{-1} -2 a^{-1} z^{-1} -2 a^{-3} z^{-1} - a^{-5} z^{-1} (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 \
-4-3-2-101234567χ
16           1-1
14          3 3
12         61 -5
10        103  7
8       116   -5
6      1410    4
4     1212     0
2    1013      -3
0   713       6
-2  39        -6
-4 17         6
-6 3          -3
-81           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{14}
r=3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=7 {\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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L11a222.gif

L11a222

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L11a224