L11a153

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L11a152.gif

L11a152

L11a154.gif

L11a154

Contents

L11a153.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a153 at Knotilus!


Link Presentations

[edit Notes on L11a153's Link Presentations]

Planar diagram presentation X8192 X10,3,11,4 X20,16,21,15 X14,5,15,6 X4,13,5,14 X22,18,7,17 X16,22,17,21 X12,20,13,19 X18,12,19,11 X2738 X6,9,1,10
Gauss code {1, -10, 2, -5, 4, -11}, {10, -1, 11, -2, 9, -8, 5, -4, 3, -7, 6, -9, 8, -3, 7, -6}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a153 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u^2 v^5-2 u^2 v^4+3 u^2 v^3-3 u^2 v^2+2 u^2 v+u v^6-3 u v^5+6 u v^4-7 u v^3+6 u v^2-3 u v+u+2 v^5-3 v^4+3 v^3-2 v^2+v}{u v^3} (db)
Jones polynomial -3 q^{9/2}+\frac{2}{q^{9/2}}+7 q^{7/2}-\frac{6}{q^{7/2}}-11 q^{5/2}+\frac{9}{q^{5/2}}+14 q^{3/2}-\frac{13}{q^{3/2}}+q^{11/2}-\frac{1}{q^{11/2}}-16 \sqrt{q}+\frac{15}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -a z^7-z^7 a^{-1} +a^3 z^5-5 a z^5-4 z^5 a^{-1} +z^5 a^{-3} +4 a^3 z^3-11 a z^3-5 z^3 a^{-1} +3 z^3 a^{-3} +6 a^3 z-12 a z+3 z a^{-3} +3 a^3 z^{-1} -5 a z^{-1} +2 a^{-1} z^{-1} (db)
Kauffman polynomial -a^2 z^{10}-z^{10}-3 a^3 z^9-7 a z^9-4 z^9 a^{-1} -2 a^4 z^8-5 a^2 z^8-7 z^8 a^{-2} -10 z^8-a^5 z^7+11 a^3 z^7+19 a z^7-z^7 a^{-1} -8 z^7 a^{-3} +7 a^4 z^6+26 a^2 z^6+8 z^6 a^{-2} -6 z^6 a^{-4} +33 z^6+5 a^5 z^5-16 a^3 z^5-22 a z^5+14 z^5 a^{-1} +12 z^5 a^{-3} -3 z^5 a^{-5} -6 a^4 z^4-33 a^2 z^4+7 z^4 a^{-4} -z^4 a^{-6} -35 z^4-8 a^5 z^3+17 a^3 z^3+29 a z^3-8 z^3 a^{-1} -10 z^3 a^{-3} +2 z^3 a^{-5} +19 a^2 z^2-3 z^2 a^{-2} -5 z^2 a^{-4} +z^2 a^{-6} +22 z^2+4 a^5 z-12 a^3 z-21 a z-2 z a^{-1} +3 z a^{-3} -5 a^2+ a^{-4} -5+3 a^3 z^{-1} +5 a z^{-1} +2 a^{-1} 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 \
-6-5-4-3-2-1012345χ
12           1-1
10          2 2
8         51 -4
6        62  4
4       85   -3
2      86    2
0     89     1
-2    57      -2
-4   48       4
-6  25        -3
-8  4         4
-1012          -1
-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} {\mathbb Z}
r=-5 {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
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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L11a152

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L11a154