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

Visit L11a8 at Knotilus!

Link Presentations

[edit Notes on L11a8's Link Presentations]

Planar diagram presentation X6172 X16,7,17,8 X4,17,1,18 X20,9,21,10 X8493 X18,22,19,21 X14,12,15,11 X12,5,13,6 X22,13,5,14 X10,19,11,20 X2,16,3,15
Gauss code {1, -11, 5, -3}, {8, -1, 2, -5, 4, -10, 7, -8, 9, -7, 11, -2, 3, -6, 10, -4, 6, -9}
A Braid Representative
A Morse Link Presentation L11a8 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1) \left(t(2)^4-5 t(2)^3+9 t(2)^2-5 t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial -q^{7/2}+5 q^{5/2}-11 q^{3/2}+17 \sqrt{q}-\frac{25}{\sqrt{q}}+\frac{27}{q^{3/2}}-\frac{27}{q^{5/2}}+\frac{23}{q^{7/2}}-\frac{17}{q^{9/2}}+\frac{10}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{1}{q^{15/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -z a^7+3 z^3 a^5+2 z a^5-a^5 z^{-1} -3 z^5 a^3-5 z^3 a^3+3 a^3 z^{-1} +z^7 a+3 z^5 a+4 z^3 a-z a-2 a z^{-1} -z^5 a^{-1} -z^3 a^{-1} (db)
Kauffman polynomial a^8 z^6-2 a^8 z^4+a^8 z^2+4 a^7 z^7-8 a^7 z^5+6 a^7 z^3-2 a^7 z+8 a^6 z^8-17 a^6 z^6+14 a^6 z^4-6 a^6 z^2+a^6+7 a^5 z^9-3 a^5 z^7-18 a^5 z^5+18 a^5 z^3-4 a^5 z-a^5 z^{-1} +2 a^4 z^{10}+22 a^4 z^8-59 a^4 z^6+47 a^4 z^4-16 a^4 z^2+3 a^4+15 a^3 z^9-8 a^3 z^7-35 a^3 z^5+z^5 a^{-3} +30 a^3 z^3-a^3 z-3 a^3 z^{-1} +2 a^2 z^{10}+27 a^2 z^8-61 a^2 z^6+5 z^6 a^{-2} +39 a^2 z^4-4 z^4 a^{-2} -11 a^2 z^2+3 a^2+8 a z^9+10 a z^7+11 z^7 a^{-1} -41 a z^5-15 z^5 a^{-1} +25 a z^3+7 z^3 a^{-1} +a z-2 a z^{-1} +13 z^8-15 z^6+4 z^4-2 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 \
8           11
6          4 -4
4         71 6
2        104  -6
0       157   8
-2      1412    -2
-4     1313     0
-6    1014      4
-8   713       -6
-10  310        7
-12 17         -6
-14 3          3
-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}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-3 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-2 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=-1 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{14}
r=0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{15}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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

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See/edit the Link Page master template (intermediate).

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