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

Visit L11n411 at Knotilus!

Link Presentations

[edit Notes on L11n411's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X13,22,14,19 X7,20,8,21 X19,10,20,11 X9,16,10,17 X17,14,18,15 X15,8,16,9 X21,18,22,5 X2536 X4,11,1,12
Gauss code {1, -10, 2, -11}, {-5, 4, -9, 3}, {10, -1, -4, 8, -6, 5, 11, -2, -3, 7, -8, 6, -7, 9}
A Braid Representative
A Morse Link Presentation L11n411 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(w-1) (w+1) \left(2 u v w^2-2 u v w+u v+w^3-2 w^2+2 w\right)}{\sqrt{u} \sqrt{v} w^{5/2}} (db)
Jones polynomial - q^{-13} +2 q^{-12} -2 q^{-11} + q^{-10} + q^{-8} +2 q^{-7} - q^{-6} +2 q^{-5} - q^{-4} + q^{-3} (db)
Signature -5 (db)
HOMFLY-PT polynomial -a^{14} z^{-2} +4 a^{12} z^{-2} +5 a^{12}-z^4 a^{10}-8 z^2 a^{10}-5 a^{10} z^{-2} -12 a^{10}+z^6 a^8+5 z^4 a^8+6 z^2 a^8+2 a^8 z^{-2} +5 a^8+z^6 a^6+5 z^4 a^6+6 z^2 a^6+2 a^6 (db)
Kauffman polynomial a^{15} z^7-5 a^{15} z^5+6 a^{15} z^3-4 a^{15} z+a^{15} z^{-1} +2 a^{14} z^8-11 a^{14} z^6+15 a^{14} z^4-8 a^{14} z^2-a^{14} z^{-2} +3 a^{14}+a^{13} z^9-3 a^{13} z^7-10 a^{13} z^5+30 a^{13} z^3-21 a^{13} z+5 a^{13} z^{-1} +4 a^{12} z^8-26 a^{12} z^6+47 a^{12} z^4-35 a^{12} z^2-4 a^{12} z^{-2} +16 a^{12}+a^{11} z^9-2 a^{11} z^7-20 a^{11} z^5+54 a^{11} z^3-39 a^{11} z+9 a^{11} z^{-1} +3 a^{10} z^8-21 a^{10} z^6+44 a^{10} z^4-43 a^{10} z^2-5 a^{10} z^{-2} +21 a^{10}+3 a^9 z^7-19 a^9 z^5+32 a^9 z^3-22 a^9 z+5 a^9 z^{-1} +a^8 z^8-5 a^8 z^6+7 a^8 z^4-10 a^8 z^2-2 a^8 z^{-2} +7 a^8+a^7 z^7-4 a^7 z^5+2 a^7 z^3+a^6 z^6-5 a^6 z^4+6 a^6 z^2-2 a^6 (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 \
-5           11
-7          110
-9        12  1
-11       111  1
-13      252   1
-15     223    3
-17    252     1
-19   221      1
-21  133       -1
-23 11         0
-25 1          1
-271           -1
Integral Khovanov Homology

(db, data source)

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