L11a198

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L11a197.gif

L11a197

L11a199.gif

L11a199

Contents

L11a198.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a198 at Knotilus!


Link Presentations

[edit Notes on L11a198's Link Presentations]

Planar diagram presentation X8192 X10,4,11,3 X22,10,7,9 X2738 X20,15,21,16 X6,14,1,13 X18,11,19,12 X16,6,17,5 X12,17,13,18 X4,20,5,19 X14,21,15,22
Gauss code {1, -4, 2, -10, 8, -6}, {4, -1, 3, -2, 7, -9, 6, -11, 5, -8, 9, -7, 10, -5, 11, -3}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a198 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1)^2 t(2)^4-t(1) t(2)^4-5 t(1)^2 t(2)^3+8 t(1) t(2)^3-4 t(2)^3+8 t(1)^2 t(2)^2-15 t(1) t(2)^2+8 t(2)^2-4 t(1)^2 t(2)+8 t(1) t(2)-5 t(2)-t(1)+1}{t(1) t(2)^2} (db)
Jones polynomial q^{11/2}-4 q^{9/2}+9 q^{7/2}-15 q^{5/2}+19 q^{3/2}-23 \sqrt{q}+\frac{21}{\sqrt{q}}-\frac{19}{q^{3/2}}+\frac{14}{q^{5/2}}-\frac{8}{q^{7/2}}+\frac{4}{q^{9/2}}-\frac{1}{q^{11/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^7 a^{-1} +3 a z^5-4 z^5 a^{-1} +z^5 a^{-3} -3 a^3 z^3+8 a z^3-8 z^3 a^{-1} +2 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 z^4 a^{-6} +a^5 z^7-3 a^5 z^5+4 z^5 a^{-5} +3 a^5 z^3-z^3 a^{-5} -a^5 z+4 a^4 z^8-14 a^4 z^6+9 z^6 a^{-4} +16 a^4 z^4-7 z^4 a^{-4} -5 a^4 z^2+2 z^2 a^{-4} -a^4+5 a^3 z^9-12 a^3 z^7+14 z^7 a^{-3} +2 a^3 z^5-20 z^5 a^{-3} +9 a^3 z^3+13 z^3 a^{-3} -4 a^3 z-4 z a^{-3} +a^3 z^{-1} +2 a^2 z^{10}+10 a^2 z^8+14 z^8 a^{-2} -46 a^2 z^6-20 z^6 a^{-2} +45 a^2 z^4+6 z^4 a^{-2} -9 a^2 z^2-3 a^2+13 a z^9+8 z^9 a^{-1} -24 a z^7+3 z^7 a^{-1} -10 a z^5-39 z^5 a^{-1} +27 a z^3+35 z^3 a^{-1} -11 a z-12 z a^{-1} +3 a z^{-1} +2 a^{-1} z^{-1} +2 z^{10}+20 z^8-61 z^6+43 z^4-6 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
12           1-1
10          3 3
8         61 -5
6        93  6
4       117   -4
2      128    4
0     1012     2
-2    911      -2
-4   510       5
-6  39        -6
-8 15         4
-10 3          -3
-121           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2
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}^{9}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{12}
r=1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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L11a197

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L11a199