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

Visit L10a1 at Knotilus!

Link Presentations

[edit Notes on L10a1's Link Presentations]

Planar diagram presentation X6172 X14,7,15,8 X4,15,1,16 X10,6,11,5 X8493 X18,11,19,12 X20,17,5,18 X12,19,13,20 X16,10,17,9 X2,14,3,13
Gauss code {1, -10, 5, -3}, {4, -1, 2, -5, 9, -4, 6, -8, 10, -2, 3, -9, 7, -6, 8, -7}
A Braid Representative
A Morse Link Presentation L10a1 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{9}{q^{9/2}}-q^{7/2}+\frac{13}{q^{7/2}}+4 q^{5/2}-\frac{17}{q^{5/2}}-8 q^{3/2}+\frac{17}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{5}{q^{11/2}}+12 \sqrt{q}-\frac{17}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a z^7-2 a^3 z^5+4 a z^5-z^5 a^{-1} +a^5 z^3-4 a^3 z^3+6 a z^3-2 z^3 a^{-1} +a z-z a^{-1} -a^5 z^{-1} +3 a^3 z^{-1} -2 a z^{-1} (db)
Kauffman polynomial -2 a^3 z^9-2 a z^9-7 a^4 z^8-13 a^2 z^8-6 z^8-9 a^5 z^7-17 a^3 z^7-15 a z^7-7 z^7 a^{-1} -5 a^6 z^6+3 a^4 z^6+15 a^2 z^6-4 z^6 a^{-2} +3 z^6-a^7 z^5+15 a^5 z^5+40 a^3 z^5+37 a z^5+12 z^5 a^{-1} -z^5 a^{-3} +6 a^6 z^4+10 a^4 z^4+5 a^2 z^4+6 z^4 a^{-2} +7 z^4-6 a^5 z^3-23 a^3 z^3-26 a z^3-8 z^3 a^{-1} +z^3 a^{-3} -2 a^4 z^2-5 a^2 z^2-2 z^2 a^{-2} -5 z^2+a^3 z+3 a z+2 z a^{-1} -a^6-3 a^4-3 a^2+a^5 z^{-1} +3 a^3 z^{-1} +2 a 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 \
8          11
6         3 -3
4        51 4
2       73  -4
0      105   5
-2     99    0
-4    88     0
-6   59      4
-8  48       -4
-10 15        4
-12 4         -4
-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}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
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

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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