L11a190

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L11a189.gif

L11a189

L11a191.gif

L11a191

Contents

L11a190.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a190 at Knotilus!


Link Presentations

[edit Notes on L11a190's Link Presentations]

Planar diagram presentation X8192 X10,3,11,4 X14,17,15,18 X16,5,17,6 X4,15,5,16 X18,13,19,14 X22,20,7,19 X20,12,21,11 X12,22,13,21 X2738 X6,9,1,10
Gauss code {1, -10, 2, -5, 4, -11}, {10, -1, 11, -2, 8, -9, 6, -3, 5, -4, 3, -6, 7, -8, 9, -7}
A Braid Representative
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A Morse Link Presentation L11a190 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 t(1) t(2)^4-t(2)^4+2 t(1)^2 t(2)^3-6 t(1) t(2)^3+4 t(2)^3-4 t(1)^2 t(2)^2+9 t(1) t(2)^2-4 t(2)^2+4 t(1)^2 t(2)-6 t(1) t(2)+2 t(2)-t(1)^2+2 t(1)}{t(1) t(2)^2} (db)
Jones polynomial q^{5/2}-3 q^{3/2}+6 \sqrt{q}-\frac{10}{\sqrt{q}}+\frac{13}{q^{3/2}}-\frac{15}{q^{5/2}}+\frac{14}{q^{7/2}}-\frac{13}{q^{9/2}}+\frac{9}{q^{11/2}}-\frac{6}{q^{13/2}}+\frac{3}{q^{15/2}}-\frac{1}{q^{17/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial z^3 a^7+z a^7-z^5 a^5-z^3 a^5+z a^5+2 a^5 z^{-1} -2 z^5 a^3-5 z^3 a^3-6 z a^3-3 a^3 z^{-1} -z^5 a-z^3 a+a z^{-1} +z^3 a^{-1} +z a^{-1} (db)
Kauffman polynomial a^9 z^7-4 a^9 z^5+5 a^9 z^3-2 a^9 z+3 a^8 z^8-12 a^8 z^6+14 a^8 z^4-4 a^8 z^2+3 a^7 z^9-8 a^7 z^7+a^7 z^5+6 a^7 z^3-a^7 z+a^6 z^{10}+5 a^6 z^8-24 a^6 z^6+23 a^6 z^4-6 a^6 z^2+6 a^5 z^9-11 a^5 z^7-6 a^5 z^5+15 a^5 z^3-9 a^5 z+2 a^5 z^{-1} +a^4 z^{10}+7 a^4 z^8-18 a^4 z^6+4 a^4 z^4+6 a^4 z^2-3 a^4+3 a^3 z^9+4 a^3 z^7-20 a^3 z^5+20 a^3 z^3-12 a^3 z+3 a^3 z^{-1} +5 a^2 z^8-a^2 z^6-11 a^2 z^4+z^4 a^{-2} +12 a^2 z^2-z^2 a^{-2} -3 a^2+6 a z^7-6 a z^5+3 z^5 a^{-1} +3 a z^3-3 z^3 a^{-1} -a z+z a^{-1} +a z^{-1} +5 z^6-5 z^4+3 z^2-1 (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 \
-8-7-6-5-4-3-2-10123χ
6           1-1
4          2 2
2         41 -3
0        62  4
-2       85   -3
-4      75    2
-6     78     1
-8    67      -1
-10   37       4
-12  36        -3
-14 14         3
-16 2          -2
-181           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-8 {\mathbb Z}
r=-7 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-5 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=3 {\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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L11a189.gif

L11a189

L11a191.gif

L11a191