L11a170

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L11a169.gif

L11a169

L11a171.gif

L11a171

Contents

L11a170.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a170 at Knotilus!


Link Presentations

[edit Notes on L11a170's Link Presentations]

Planar diagram presentation X8192 X10,4,11,3 X12,7,13,8 X22,15,7,16 X14,6,15,5 X6,14,1,13 X16,21,17,22 X18,10,19,9 X20,11,21,12 X4,18,5,17 X2,19,3,20
Gauss code {1, -11, 2, -10, 5, -6}, {3, -1, 8, -2, 9, -3, 6, -5, 4, -7, 10, -8, 11, -9, 7, -4}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a170 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u^2 v^4-5 u^2 v^3+8 u^2 v^2-6 u^2 v+2 u^2-3 u v^4+11 u v^3-15 u v^2+11 u v-3 u+2 v^4-6 v^3+8 v^2-5 v+1}{u v^2} (db)
Jones polynomial q^{9/2}-\frac{10}{q^{9/2}}-5 q^{7/2}+\frac{18}{q^{7/2}}+11 q^{5/2}-\frac{24}{q^{5/2}}-19 q^{3/2}+\frac{28}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{4}{q^{11/2}}+24 \sqrt{q}-\frac{29}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a z^7-2 a^3 z^5+2 a z^5-2 z^5 a^{-1} +a^5 z^3-3 a^3 z^3+a z^3-2 z^3 a^{-1} +z^3 a^{-3} +a^5 z-a^3 z-a z+2 z a^{-1} + a^{-1} z^{-1} - a^{-3} z^{-1} (db)
Kauffman polynomial a^7 z^5-a^7 z^3+4 a^6 z^6-4 a^6 z^4+a^6 z^2+9 a^5 z^7-12 a^5 z^5+7 a^5 z^3-a^5 z+13 a^4 z^8-21 a^4 z^6+z^6 a^{-4} +17 a^4 z^4-z^4 a^{-4} -6 a^4 z^2+11 a^3 z^9-11 a^3 z^7+5 z^7 a^{-3} -9 z^5 a^{-3} +2 a^3 z^3+4 z^3 a^{-3} +a^3 z+z a^{-3} - a^{-3} z^{-1} +4 a^2 z^{10}+17 a^2 z^8+10 z^8 a^{-2} -48 a^2 z^6-20 z^6 a^{-2} +38 a^2 z^4+10 z^4 a^{-2} -12 a^2 z^2-z^2 a^{-2} + a^{-2} +21 a z^9+10 z^9 a^{-1} -39 a z^7-14 z^7 a^{-1} +21 a z^5-z^5 a^{-1} -10 a z^3+5 a z+4 z a^{-1} - a^{-1} z^{-1} +4 z^{10}+14 z^8-44 z^6+28 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 \
-6-5-4-3-2-1012345χ
10           1-1
8          4 4
6         71 -6
4        124  8
2       138   -5
0      1611    5
-2     1314     1
-4    1115      -4
-6   713       6
-8  311        -8
-10 17         6
-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}^{7}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-2 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=-1 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=0 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{15} {\mathbb Z}^{16}
r=1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{12}
r=3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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L11a169

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L11a171