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

Visit L11a12 at Knotilus!

Link Presentations

[edit Notes on L11a12's Link Presentations]

Planar diagram presentation X6172 X20,7,21,8 X4,21,1,22 X14,12,15,11 X8493 X12,5,13,6 X22,13,5,14 X18,15,19,16 X16,9,17,10 X10,17,11,18 X2,20,3,19
Gauss code {1, -11, 5, -3}, {6, -1, 2, -5, 9, -10, 4, -6, 7, -4, 8, -9, 10, -8, 11, -2, 3, -7}
A Braid Representative
A Morse Link Presentation L11a12 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u-1) (v-1) \left(v^4-5 v^3+7 v^2-5 v+1\right)}{\sqrt{u} v^{5/2}} (db)
Jones polynomial q^{3/2}-5 \sqrt{q}+\frac{10}{\sqrt{q}}-\frac{17}{q^{3/2}}+\frac{21}{q^{5/2}}-\frac{25}{q^{7/2}}+\frac{24}{q^{9/2}}-\frac{21}{q^{11/2}}+\frac{15}{q^{13/2}}-\frac{8}{q^{15/2}}+\frac{4}{q^{17/2}}-\frac{1}{q^{19/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^9 z-3 a^7 z^3-4 a^7 z-a^7 z^{-1} +3 a^5 z^5+7 a^5 z^3+6 a^5 z+2 a^5 z^{-1} -a^3 z^7-3 a^3 z^5-4 a^3 z^3-2 a^3 z+a z^5+a z^3-a z-a z^{-1} (db)
Kauffman polynomial a^{11} z^5-a^{11} z^3+4 a^{10} z^6-6 a^{10} z^4+3 a^{10} z^2+7 a^9 z^7-9 a^9 z^5+5 a^9 z^3-2 a^9 z+8 a^8 z^8-5 a^8 z^6-7 a^8 z^4+9 a^8 z^2-2 a^8+6 a^7 z^9+5 a^7 z^7-25 a^7 z^5+24 a^7 z^3-8 a^7 z+a^7 z^{-1} +2 a^6 z^{10}+17 a^6 z^8-36 a^6 z^6+15 a^6 z^4+8 a^6 z^2-5 a^6+13 a^5 z^9-10 a^5 z^7-26 a^5 z^5+31 a^5 z^3-11 a^5 z+2 a^5 z^{-1} +2 a^4 z^{10}+18 a^4 z^8-47 a^4 z^6+27 a^4 z^4-3 a^4+7 a^3 z^9-3 a^3 z^7-21 a^3 z^5+18 a^3 z^3-4 a^3 z+9 a^2 z^8-19 a^2 z^6+10 a^2 z^4-2 a^2 z^2+a^2+5 a z^7-10 a z^5+5 a z^3+a z-a z^{-1} +z^6-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          4 4
0         61 -5
-2        114  7
-4       128   -4
-6      139    4
-8     1112     1
-10    1013      -3
-12   511       6
-14  310        -7
-16 15         4
-18 3          -3
-201           1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-4 i=-2
r=-8 {\mathbb Z}
r=-7 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-5 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-4 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-3 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=-2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{11}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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

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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