L11a195

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L11a194.gif

L11a194

L11a196.gif

L11a196

Contents

L11a195.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a195 at Knotilus!


Link Presentations

[edit Notes on L11a195's Link Presentations]

Planar diagram presentation X8192 X10,4,11,3 X22,10,7,9 X2738 X18,13,19,14 X6,12,1,11 X20,15,21,16 X14,19,15,20 X4,18,5,17 X16,6,17,5 X12,21,13,22
Gauss code {1, -4, 2, -9, 10, -6}, {4, -1, 3, -2, 6, -11, 5, -8, 7, -10, 9, -5, 8, -7, 11, -3}
A Braid Representative
BraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a195 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u^2 v^4-3 u^2 v^3+4 u^2 v^2-4 u^2 v+2 u^2-3 u v^4+8 u v^3-9 u v^2+8 u v-3 u+2 v^4-4 v^3+4 v^2-3 v+1}{u v^2} (db)
Jones polynomial q^{9/2}-3 q^{7/2}+6 q^{5/2}-12 q^{3/2}+15 \sqrt{q}-\frac{19}{\sqrt{q}}+\frac{19}{q^{3/2}}-\frac{17}{q^{5/2}}+\frac{13}{q^{7/2}}-\frac{8}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{1}{q^{13/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a z^7-2 a^3 z^5+4 a z^5-2 z^5 a^{-1} +a^5 z^3-5 a^3 z^3+8 a z^3-6 z^3 a^{-1} +z^3 a^{-3} +a^5 z-4 a^3 z+8 a z-6 z a^{-1} +2 z a^{-3} -a^3 z^{-1} +3 a z^{-1} -2 a^{-1} z^{-1} (db)
Kauffman polynomial a^7 z^5-a^7 z^3+4 a^6 z^6-6 a^6 z^4+2 a^6 z^2+7 a^5 z^7-11 a^5 z^5+5 a^5 z^3-a^5 z+7 a^4 z^8-7 a^4 z^6+z^6 a^{-4} -2 a^4 z^4-3 z^4 a^{-4} +3 a^4 z^2+2 z^2 a^{-4} -a^4+4 a^3 z^9+5 a^3 z^7+3 z^7 a^{-3} -19 a^3 z^5-9 z^5 a^{-3} +14 a^3 z^3+9 z^3 a^{-3} -4 a^3 z-4 z a^{-3} +a^3 z^{-1} +a^2 z^{10}+12 a^2 z^8+4 z^8 a^{-2} -24 a^2 z^6-8 z^6 a^{-2} +12 a^2 z^4+2 z^4 a^{-2} +3 a^2 z^2+2 z^2 a^{-2} -3 a^2+7 a z^9+3 z^9 a^{-1} -4 a z^7+z^7 a^{-1} -15 a z^5-17 z^5 a^{-1} +21 a z^3+22 z^3 a^{-1} -11 a z-12 z a^{-1} +3 a z^{-1} +2 a^{-1} z^{-1} +z^{10}+9 z^8-22 z^6+13 z^4+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 \
-6-5-4-3-2-1012345χ
10           1-1
8          2 2
6         41 -3
4        82  6
2       85   -3
0      117    4
-2     99     0
-4    810      -2
-6   59       4
-8  38        -5
-10 15         4
-12 3          -3
-141           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{11}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=5 {\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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L11a194

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L11a196