L11a183

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L11a182.gif

L11a182

L11a184.gif

L11a184

Contents

L11a183.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a183 at Knotilus!


Link Presentations

[edit Notes on L11a183's Link Presentations]

Planar diagram presentation X8192 X10,4,11,3 X18,7,19,8 X20,10,21,9 X22,17,7,18 X14,6,15,5 X16,12,17,11 X12,16,13,15 X4,14,5,13 X6,20,1,19 X2,21,3,22
Gauss code {1, -11, 2, -9, 6, -10}, {3, -1, 4, -2, 7, -8, 9, -6, 8, -7, 5, -3, 10, -4, 11, -5}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation L11a183 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 t(1)^2 t(2)^4-3 t(1) t(2)^4+t(2)^4-5 t(1)^2 t(2)^3+9 t(1) t(2)^3-4 t(2)^3+6 t(1)^2 t(2)^2-13 t(1) t(2)^2+6 t(2)^2-4 t(1)^2 t(2)+9 t(1) t(2)-5 t(2)+t(1)^2-3 t(1)+2}{t(1) t(2)^2} (db)
Jones polynomial -15 q^{9/2}+20 q^{7/2}-\frac{1}{q^{7/2}}-24 q^{5/2}+\frac{4}{q^{5/2}}+23 q^{3/2}-\frac{9}{q^{3/2}}+q^{15/2}-4 q^{13/2}+9 q^{11/2}-21 \sqrt{q}+\frac{15}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial z^5 a^{-5} +2 z^3 a^{-5} +z a^{-5} -z^7 a^{-3} -3 z^5 a^{-3} -3 z^3 a^{-3} -z a^{-3} - a^{-3} z^{-1} -z^7 a^{-1} +a z^5-3 z^5 a^{-1} +2 a z^3-3 z^3 a^{-1} +a z+ a^{-1} z^{-1} (db)
Kauffman polynomial z^6 a^{-8} -2 z^4 a^{-8} +z^2 a^{-8} +4 z^7 a^{-7} -9 z^5 a^{-7} +6 z^3 a^{-7} -z a^{-7} +7 z^8 a^{-6} -15 z^6 a^{-6} +9 z^4 a^{-6} -z^2 a^{-6} +6 z^9 a^{-5} -5 z^7 a^{-5} -9 z^5 a^{-5} +8 z^3 a^{-5} -z a^{-5} +2 z^{10} a^{-4} +13 z^8 a^{-4} -33 z^6 a^{-4} +19 z^4 a^{-4} -3 z^2 a^{-4} +12 z^9 a^{-3} -12 z^7 a^{-3} +a^3 z^5-10 z^5 a^{-3} -a^3 z^3+9 z^3 a^{-3} - a^{-3} z^{-1} +2 z^{10} a^{-2} +15 z^8 a^{-2} +4 a^2 z^6-30 z^6 a^{-2} -5 a^2 z^4+13 z^4 a^{-2} +2 a^2 z^2-z^2 a^{-2} + a^{-2} +6 z^9 a^{-1} +8 a z^7+5 z^7 a^{-1} -12 a z^5-23 z^5 a^{-1} +8 a z^3+16 z^3 a^{-1} -2 a z-2 z a^{-1} - a^{-1} z^{-1} +9 z^8-9 z^6+2 z^2 (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        93  6
8       116   -5
6      139    4
4     1112     1
2    1012      -2
0   612       6
-2  39        -6
-4 16         5
-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}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{13}
r=3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
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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