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

Visit L10a6 at Knotilus!

Link Presentations

[edit Notes on L10a6's Link Presentations]

Planar diagram presentation X6172 X18,7,19,8 X4,19,1,20 X12,6,13,5 X8493 X16,13,17,14 X14,9,15,10 X10,15,11,16 X20,12,5,11 X2,18,3,17
Gauss code {1, -10, 5, -3}, {4, -1, 2, -5, 7, -8, 9, -4, 6, -7, 8, -6, 10, -2, 3, -9}
A Braid Representative
A Morse Link Presentation L10a6 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1) \left(v^4-3 v^3+5 v^2-3 v+1\right)}{\sqrt{u} v^{5/2}} (db)
Jones polynomial -\frac{8}{q^{9/2}}-q^{7/2}+\frac{13}{q^{7/2}}+4 q^{5/2}-\frac{17}{q^{5/2}}-9 q^{3/2}+\frac{17}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{4}{q^{11/2}}+13 \sqrt{q}-\frac{17}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^5 z^3+a^5 z-2 a^3 z^5-5 a^3 z^3-4 a^3 z+a z^7+4 a z^5-z^5 a^{-1} +7 a z^3-2 z^3 a^{-1} +5 a z-2 z a^{-1} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial a^7 z^5-a^7 z^3+4 a^6 z^6-6 a^6 z^4+3 a^6 z^2+7 a^5 z^7-11 a^5 z^5+7 a^5 z^3-2 a^5 z+6 a^4 z^8-a^4 z^6-12 a^4 z^4+8 a^4 z^2+2 a^3 z^9+16 a^3 z^7-39 a^3 z^5+z^5 a^{-3} +28 a^3 z^3-z^3 a^{-3} -8 a^3 z+13 a^2 z^8-14 a^2 z^6+4 z^6 a^{-2} -7 a^2 z^4-5 z^4 a^{-2} +7 a^2 z^2+2 z^2 a^{-2} +2 a z^9+17 a z^7+8 z^7 a^{-1} -42 a z^5-14 z^5 a^{-1} +31 a z^3+10 z^3 a^{-1} -10 a z-4 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +7 z^8-5 z^6-6 z^4+4 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          11
6         3 -3
4        61 5
2       73  -4
0      106   4
-2     99    0
-4    88     0
-6   59      4
-8  38       -5
-10 15        4
-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}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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