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

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Link Presentations

[edit Notes on L11a57's Link Presentations]

Planar diagram presentation X6172 X12,4,13,3 X14,8,15,7 X18,16,19,15 X16,9,17,10 X8,17,9,18 X22,20,5,19 X20,11,21,12 X10,21,11,22 X2536 X4,14,1,13
Gauss code {1, -10, 2, -11}, {10, -1, 3, -6, 5, -9, 8, -2, 11, -3, 4, -5, 6, -4, 7, -8, 9, -7}
A Braid Representative
A Morse Link Presentation L11a57 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1)-1) (t(2)-1) \left(t(2)^2-3 t(2)+1\right) \left(t(2)^2-t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial -4 q^{9/2}+\frac{3}{q^{9/2}}+8 q^{7/2}-\frac{7}{q^{7/2}}-13 q^{5/2}+\frac{12}{q^{5/2}}+17 q^{3/2}-\frac{16}{q^{3/2}}+q^{11/2}-\frac{1}{q^{11/2}}-20 \sqrt{q}+\frac{18}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial a^5 z+a^5 z^{-1} +z^5 a^{-3} -3 a^3 z^3+2 z^3 a^{-3} -6 a^3 z+z a^{-3} -3 a^3 z^{-1} -z^7 a^{-1} +3 a z^5-4 z^5 a^{-1} +9 a z^3-7 z^3 a^{-1} +9 a z-5 z a^{-1} +4 a z^{-1} -2 a^{-1} z^{-1} (db)
Kauffman polynomial -a^2 z^{10}-z^{10}-3 a^3 z^9-8 a z^9-5 z^9 a^{-1} -3 a^4 z^8-11 a^2 z^8-10 z^8 a^{-2} -18 z^8-a^5 z^7+3 a^3 z^7+5 a z^7-10 z^7 a^{-1} -11 z^7 a^{-3} +11 a^4 z^6+44 a^2 z^6+10 z^6 a^{-2} -8 z^6 a^{-4} +51 z^6+4 a^5 z^5+17 a^3 z^5+42 a z^5+47 z^5 a^{-1} +14 z^5 a^{-3} -4 z^5 a^{-5} -14 a^4 z^4-47 a^2 z^4+3 z^4 a^{-2} +7 z^4 a^{-4} -z^4 a^{-6} -38 z^4-6 a^5 z^3-32 a^3 z^3-60 a z^3-44 z^3 a^{-1} -8 z^3 a^{-3} +2 z^3 a^{-5} +7 a^4 z^2+19 a^2 z^2-2 z^2 a^{-2} -2 z^2 a^{-4} +12 z^2+4 a^5 z+18 a^3 z+26 a z+15 z a^{-1} +3 z a^{-3} -a^4-3 a^2- a^{-2} -2-a^5 z^{-1} -3 a^3 z^{-1} -4 a z^{-1} -2 a^{-1} 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 \
12           1-1
10          3 3
8         51 -4
6        83  5
4       95   -4
2      118    3
0     911     2
-2    79      -2
-4   59       4
-6  27        -5
-8 15         4
-10 2          -2
-121           1
Integral Khovanov Homology

(db, data source)

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