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

Visit L10a98 at Knotilus!

Link Presentations

[edit Notes on L10a98's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X14,5,15,6 X18,7,19,8 X20,15,9,16 X16,19,17,20 X8,9,1,10 X4,13,5,14 X6,17,7,18
Gauss code {1, -2, 3, -9, 4, -10, 5, -8}, {8, -1, 2, -3, 9, -4, 6, -7, 10, -5, 7, -6}
A Braid Representative
A Morse Link Presentation L10a98 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 t(2)^3 t(1)^3-t(2)^2 t(1)^3-t(2)^3 t(1)^2+3 t(2)^2 t(1)^2-t(2) t(1)^2-t(2)^2 t(1)+3 t(2) t(1)-t(1)-t(2)+2}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial \frac{1}{q^{9/2}}-\frac{1}{q^{7/2}}-\frac{1}{q^{27/2}}+\frac{2}{q^{25/2}}-\frac{3}{q^{23/2}}+\frac{4}{q^{21/2}}-\frac{5}{q^{19/2}}+\frac{5}{q^{17/2}}-\frac{4}{q^{15/2}}+\frac{3}{q^{13/2}}-\frac{3}{q^{11/2}} (db)
Signature -7 (db)
HOMFLY-PT polynomial a^{11} z^5+4 a^{11} z^3+3 a^{11} z-a^9 z^7-5 a^9 z^5-6 a^9 z^3+a^9 z^{-1} -a^7 z^7-6 a^7 z^5-11 a^7 z^3-7 a^7 z-a^7 z^{-1} (db)
Kauffman polynomial -z^3 a^{17}+z a^{17}-2 z^4 a^{16}+2 z^2 a^{16}-2 z^5 a^{15}+z^3 a^{15}-2 z^6 a^{14}+2 z^4 a^{14}-z^2 a^{14}-2 z^7 a^{13}+4 z^5 a^{13}-3 z^3 a^{13}+z a^{13}-2 z^8 a^{12}+7 z^6 a^{12}-9 z^4 a^{12}+5 z^2 a^{12}-z^9 a^{11}+3 z^7 a^{11}-3 z^5 a^{11}+5 z^3 a^{11}-3 z a^{11}-3 z^8 a^{10}+13 z^6 a^{10}-15 z^4 a^{10}+5 z^2 a^{10}-z^9 a^9+4 z^7 a^9-3 z^5 a^9-z^3 a^9+2 z a^9-a^9 z^{-1} -z^8 a^8+4 z^6 a^8-2 z^4 a^8-3 z^2 a^8+a^8-z^7 a^7+6 z^5 a^7-11 z^3 a^7+7 z a^7-a^7 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          11
-8         110
-10        2  2
-12       11  0
-14      32   1
-16     21    -1
-18    33     0
-20   12      1
-22  23       -1
-24 12        1
-26 1         -1
-281          1
Integral Khovanov Homology

(db, data source)

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