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

Visit L10a23 at Knotilus!

Link Presentations

[edit Notes on L10a23's Link Presentations]

Planar diagram presentation X6172 X16,7,17,8 X18,9,19,10 X8,17,9,18 X4,19,1,20 X14,12,15,11 X10,4,11,3 X12,5,13,6 X20,13,5,14 X2,16,3,15
Gauss code {1, -10, 7, -5}, {8, -1, 2, -4, 3, -7, 6, -8, 9, -6, 10, -2, 4, -3, 5, -9}
A Braid Representative
A Morse Link Presentation L10a23 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u-1) (v-1) \left(v^2+1\right) \left(v^2-3 v+1\right)}{\sqrt{u} v^{5/2}} (db)
Jones polynomial q^{3/2}-4 \sqrt{q}+\frac{6}{\sqrt{q}}-\frac{10}{q^{3/2}}+\frac{12}{q^{5/2}}-\frac{14}{q^{7/2}}+\frac{12}{q^{9/2}}-\frac{10}{q^{11/2}}+\frac{7}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{1}{q^{17/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^7 \left(-z^3\right)-2 a^7 z-a^7 z^{-1} +2 a^5 z^5+6 a^5 z^3+5 a^5 z+2 a^5 z^{-1} -a^3 z^7-4 a^3 z^5-5 a^3 z^3-2 a^3 z+a z^5+2 a z^3-a z-a z^{-1} (db)
Kauffman polynomial a^{10} z^4-a^{10} z^2+3 a^9 z^5-2 a^9 z^3+6 a^8 z^6-8 a^8 z^4+6 a^8 z^2-2 a^8+7 a^7 z^7-10 a^7 z^5+8 a^7 z^3-4 a^7 z+a^7 z^{-1} +5 a^6 z^8-2 a^6 z^6-9 a^6 z^4+12 a^6 z^2-5 a^6+2 a^5 z^9+7 a^5 z^7-23 a^5 z^5+20 a^5 z^3-9 a^5 z+2 a^5 z^{-1} +10 a^4 z^8-23 a^4 z^6+12 a^4 z^4+3 a^4 z^2-3 a^4+2 a^3 z^9+4 a^3 z^7-22 a^3 z^5+18 a^3 z^3-4 a^3 z+5 a^2 z^8-14 a^2 z^6+10 a^2 z^4-2 a^2 z^2+a^2+4 a z^7-12 a z^5+8 a z^3+a z-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       73  4
-4      75   -2
-6     75    2
-8    57     2
-10   57      -2
-12  25       3
-14 15        -4
-16 2         2
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{7}
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).

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