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

Visit L11a139 at Knotilus!

Link Presentations

[edit Notes on L11a139's Link Presentations]

Planar diagram presentation X8192 X10,4,11,3 X22,10,7,9 X2738 X4,15,5,16 X12,5,13,6 X16,12,17,11 X6,18,1,17 X14,20,15,19 X20,14,21,13 X18,21,19,22
Gauss code {1, -4, 2, -5, 6, -8}, {4, -1, 3, -2, 7, -6, 10, -9, 5, -7, 8, -11, 9, -10, 11, -3}
A Braid Representative
A Morse Link Presentation L11a139 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u^2 v^4-5 u^2 v^3+8 u^2 v^2-5 u^2 v+u^2-2 u v^4+10 u v^3-15 u v^2+10 u v-2 u+v^4-5 v^3+8 v^2-5 v+1}{u v^2} (db)
Jones polynomial -q^{13/2}+4 q^{11/2}-9 q^{9/2}+16 q^{7/2}-22 q^{5/2}+25 q^{3/2}-26 \sqrt{q}+\frac{22}{\sqrt{q}}-\frac{17}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{5}{q^{7/2}}+\frac{1}{q^{9/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^7 a^{-1} +2 a z^5-3 z^5 a^{-1} +2 z^5 a^{-3} -a^3 z^3+3 a z^3-5 z^3 a^{-1} +4 z^3 a^{-3} -z^3 a^{-5} -3 z a^{-1} +3 z a^{-3} -z a^{-5} +a^3 z^{-1} -a z^{-1} (db)
Kauffman polynomial z^5 a^{-7} -z^3 a^{-7} +4 z^6 a^{-6} -5 z^4 a^{-6} +2 z^2 a^{-6} +8 z^7 a^{-5} -11 z^5 a^{-5} +7 z^3 a^{-5} -2 z a^{-5} +10 z^8 a^{-4} +a^4 z^6-12 z^6 a^{-4} -a^4 z^4+5 z^4 a^{-4} +7 z^9 a^{-3} +5 a^3 z^7+4 z^7 a^{-3} -10 a^3 z^5-25 z^5 a^{-3} +5 a^3 z^3+22 z^3 a^{-3} +a^3 z-6 z a^{-3} -a^3 z^{-1} +2 z^{10} a^{-2} +9 a^2 z^8+21 z^8 a^{-2} -19 a^2 z^6-49 z^6 a^{-2} +11 a^2 z^4+34 z^4 a^{-2} -2 a^2 z^2-8 z^2 a^{-2} +a^2+7 a z^9+14 z^9 a^{-1} -2 a z^7-11 z^7 a^{-1} -22 a z^5-25 z^5 a^{-1} +17 a z^3+26 z^3 a^{-1} -a z-6 z a^{-1} -a z^{-1} +2 z^{10}+20 z^8-53 z^6+36 z^4-8 z^2 (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 \
14           11
12          3 -3
10         61 5
8        103  -7
6       126   6
4      1310    -3
2     1312     1
0    1014      4
-2   712       -5
-4  411        7
-6 16         -5
-8 4          4
-101           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=0 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{13}
r=1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=6 {\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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