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

Visit L10a2 at Knotilus!

Link Presentations

[edit Notes on L10a2's Link Presentations]

Planar diagram presentation X6172 X16,7,17,8 X4,17,1,18 X14,10,15,9 X8493 X10,5,11,6 X20,13,5,14 X18,11,19,12 X12,19,13,20 X2,16,3,15
Gauss code {1, -10, 5, -3}, {6, -1, 2, -5, 4, -6, 8, -9, 7, -4, 10, -2, 3, -8, 9, -7}
A Braid Representative
A Morse Link Presentation L10a2 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1)-1) (t(2)-1) \left(t(2)^4-3 t(2)^3+3 t(2)^2-3 t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial \frac{14}{q^{9/2}}-\frac{15}{q^{7/2}}+\frac{13}{q^{5/2}}+q^{3/2}-\frac{11}{q^{3/2}}+\frac{1}{q^{17/2}}-\frac{4}{q^{15/2}}+\frac{8}{q^{13/2}}-\frac{11}{q^{11/2}}-4 \sqrt{q}+\frac{6}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^7 \left(-z^3\right)-a^7 z+2 a^5 z^5+5 a^5 z^3+2 a^5 z-a^5 z^{-1} -a^3 z^7-4 a^3 z^5-5 a^3 z^3+3 a^3 z^{-1} +a z^5+2 a z^3-a z-2 a z^{-1} (db)
Kauffman polynomial a^{10} z^4+4 a^9 z^5-2 a^9 z^3+8 a^8 z^6-9 a^8 z^4+3 a^8 z^2+9 a^7 z^7-11 a^7 z^5+5 a^7 z^3-2 a^7 z+6 a^6 z^8-a^6 z^6-9 a^6 z^4+2 a^6 z^2+a^6+2 a^5 z^9+11 a^5 z^7-30 a^5 z^5+20 a^5 z^3-4 a^5 z-a^5 z^{-1} +11 a^4 z^8-23 a^4 z^6+12 a^4 z^4-4 a^4 z^2+3 a^4+2 a^3 z^9+6 a^3 z^7-27 a^3 z^5+22 a^3 z^3-a^3 z-3 a^3 z^{-1} +5 a^2 z^8-13 a^2 z^6+9 a^2 z^4-3 a^2 z^2+3 a^2+4 a z^7-12 a z^5+9 a z^3+a z-2 a z^{-1} +z^6-2 z^4 (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 \
4          1-1
2         3 3
0        31 -2
-2       83  5
-4      75   -2
-6     86    2
-8    67     1
-10   58      -3
-12  36       3
-14 15        -4
-16 3         3
-181          -1
Integral Khovanov Homology

(db, data source)

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

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

See/edit the Link_Splice_Base (expert).

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