L11a174

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L11a173.gif

L11a173

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L11a175

Contents

L11a174.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a174 at Knotilus!


Link Presentations

[edit Notes on L11a174's Link Presentations]

Planar diagram presentation X8192 X2,9,3,10 X10,3,11,4 X16,8,17,7 X22,16,7,15 X6,21,1,22 X20,11,21,12 X14,6,15,5 X18,13,19,14 X12,19,13,20 X4,18,5,17
Gauss code {1, -2, 3, -11, 8, -6}, {4, -1, 2, -3, 7, -10, 9, -8, 5, -4, 11, -9, 10, -7, 6, -5}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a174 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 u^2 v^4-4 u^2 v^3+4 u^2 v^2-2 u^2 v-2 u v^4+7 u v^3-11 u v^2+7 u v-2 u-2 v^3+4 v^2-4 v+2}{u v^2} (db)
Jones polynomial -\frac{11}{q^{9/2}}-q^{7/2}+\frac{14}{q^{7/2}}+3 q^{5/2}-\frac{17}{q^{5/2}}-7 q^{3/2}+\frac{17}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{3}{q^{13/2}}+\frac{6}{q^{11/2}}+11 \sqrt{q}-\frac{15}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^3 z^7+a z^7-a^5 z^5+4 a^3 z^5+4 a z^5-z^5 a^{-1} -3 a^5 z^3+5 a^3 z^3+6 a z^3-3 z^3 a^{-1} -2 a^5 z+4 a z-3 z a^{-1} +a^5 z^{-1} -2 a^3 z^{-1} +2 a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial a^8 z^6-3 a^8 z^4+2 a^8 z^2+3 a^7 z^7-9 a^7 z^5+7 a^7 z^3-a^7 z+4 a^6 z^8-9 a^6 z^6+3 a^6 z^4+a^6 z^2+4 a^5 z^9-9 a^5 z^7+9 a^5 z^5-8 a^5 z^3+a^5 z^{-1} +2 a^4 z^{10}-4 a^4 z^6+2 a^4 z^4-2 a^4 z^2+9 a^3 z^9-26 a^3 z^7+35 a^3 z^5+z^5 a^{-3} -18 a^3 z^3-2 z^3 a^{-3} -2 a^3 z+2 a^3 z^{-1} +2 a^2 z^{10}+3 a^2 z^8-14 a^2 z^6+3 z^6 a^{-2} +19 a^2 z^4-5 z^4 a^{-2} -7 a^2 z^2+a^2+5 a z^9-8 a z^7+6 z^7 a^{-1} +2 a z^5-14 z^5 a^{-1} +11 a z^3+12 z^3 a^{-1} -8 a z-5 z a^{-1} +2 a z^{-1} + a^{-1} z^{-1} +7 z^8-17 z^6+18 z^4-6 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 \
-7-6-5-4-3-2-101234χ
8           11
6          2 -2
4         51 4
2        62  -4
0       95   4
-2      97    -2
-4     88     0
-6    710      3
-8   47       -3
-10  27        5
-12 14         -3
-14 2          2
-161           -1
Integral Khovanov Homology

(db, data source)

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