L11a178

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L11a177.gif

L11a177

L11a179.gif

L11a179

Contents

L11a178.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a178 at Knotilus!


Link Presentations

[edit Notes on L11a178's Link Presentations]

Planar diagram presentation X8192 X10,4,11,3 X16,7,17,8 X20,10,21,9 X22,15,7,16 X14,6,15,5 X18,14,19,13 X12,20,13,19 X4,12,5,11 X6,18,1,17 X2,21,3,22
Gauss code {1, -11, 2, -9, 6, -10}, {3, -1, 4, -2, 9, -8, 7, -6, 5, -3, 10, -7, 8, -4, 11, -5}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a178 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 u^2 v^4-5 u^2 v^3+6 u^2 v^2-4 u^2 v+u^2-3 u v^4+10 u v^3-15 u v^2+10 u v-3 u+v^4-4 v^3+6 v^2-5 v+2}{u v^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} -z^7 a^{-3} +a z^5-3 z^5 a^{-1} -3 z^5 a^{-3} +z^5 a^{-5} +2 a z^3-4 z^3 a^{-1} -3 z^3 a^{-3} +2 z^3 a^{-5} +2 a z-3 z a^{-1} +z a^{-3} +z a^{-5} +a z^{-1} -2 a^{-1} z^{-1} +2 a^{-3} z^{-1} - a^{-5} z^{-1} (db)
Kauffman polynomial -3 z^{10} a^{-2} -3 z^{10} a^{-4} -9 z^9 a^{-1} -16 z^9 a^{-3} -7 z^9 a^{-5} -15 z^8 a^{-2} -10 z^8 a^{-4} -7 z^8 a^{-6} -12 z^8-9 a z^7+7 z^7 a^{-1} +29 z^7 a^{-3} +9 z^7 a^{-5} -4 z^7 a^{-7} -4 a^2 z^6+42 z^6 a^{-2} +32 z^6 a^{-4} +14 z^6 a^{-6} -z^6 a^{-8} +21 z^6-a^3 z^5+14 a z^5+9 z^5 a^{-1} -12 z^5 a^{-3} +3 z^5 a^{-5} +9 z^5 a^{-7} +4 a^2 z^4-34 z^4 a^{-2} -23 z^4 a^{-4} -7 z^4 a^{-6} +2 z^4 a^{-8} -16 z^4+a^3 z^3-9 a z^3-16 z^3 a^{-1} -5 z^3 a^{-3} -5 z^3 a^{-5} -6 z^3 a^{-7} +9 z^2 a^{-2} +6 z^2 a^{-4} +z^2 a^{-6} -z^2 a^{-8} +5 z^2+4 a z+9 z a^{-1} +7 z a^{-3} +3 z a^{-5} +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).

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