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

Visit L11n106 at Knotilus!

Link Presentations

[edit Notes on L11n106's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X7,16,8,17 X17,22,18,5 X13,18,14,19 X9,21,10,20 X19,14,20,15 X21,9,22,8 X15,10,16,11 X2536 X4,11,1,12
Gauss code {1, -10, 2, -11}, {10, -1, -3, 8, -6, 9, 11, -2, -5, 7, -9, 3, -4, 5, -7, 6, -8, 4}
A Braid Representative
A Morse Link Presentation L11n106 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u v^5-2 u v^4-u v^3+3 u v^2-2 u v-2 v^4+3 v^3-v^2-2 v+1}{\sqrt{u} v^{5/2}} (db)
Jones polynomial -\frac{1}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{1}{q^{5/2}}+\frac{1}{q^{9/2}}-\frac{3}{q^{11/2}}+\frac{2}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{2}{q^{19/2}}+\frac{1}{q^{21/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial -a^{11} z^{-1} +4 z a^9+3 a^9 z^{-1} -z^5 a^7-7 z^3 a^7-9 z a^7-3 a^7 z^{-1} +z^7 a^5+6 z^5 a^5+10 z^3 a^5+8 z a^5+2 a^5 z^{-1} -z^5 a^3-5 z^3 a^3-5 z a^3-a^3 z^{-1} (db)
Kauffman polynomial a^{12} z^6-4 a^{12} z^4+3 a^{12} z^2-a^{12}+2 a^{11} z^7-8 a^{11} z^5+6 a^{11} z^3-3 a^{11} z+a^{11} z^{-1} +2 a^{10} z^8-8 a^{10} z^6+4 a^{10} z^4+3 a^{10} z^2-2 a^{10}+a^9 z^9-3 a^9 z^7-5 a^9 z^5+15 a^9 z^3-11 a^9 z+3 a^9 z^{-1} +3 a^8 z^8-17 a^8 z^6+24 a^8 z^4-9 a^8 z^2+a^7 z^9-4 a^7 z^7-5 a^7 z^5+22 a^7 z^3-14 a^7 z+3 a^7 z^{-1} +2 a^6 z^8-14 a^6 z^6+25 a^6 z^4-14 a^6 z^2+2 a^6+2 a^5 z^7-14 a^5 z^5+23 a^5 z^3-12 a^5 z+2 a^5 z^{-1} +a^4 z^8-6 a^4 z^6+9 a^4 z^4-5 a^4 z^2+a^3 z^7-6 a^3 z^5+10 a^3 z^3-6 a^3 z+a^3 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 \
0           11
-2            0
-4        121 0
-6       111  1
-8      221   -1
-10     321    2
-12    241     1
-14   221      1
-16  121       0
-18 12         -1
-20 1          1
-221           -1
Integral Khovanov Homology

(db, data source)

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