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

Visit L11a20 at Knotilus!

Link Presentations

[edit Notes on L11a20's Link Presentations]

Planar diagram presentation X6172 X10,4,11,3 X12,8,13,7 X18,16,19,15 X16,9,17,10 X8,17,9,18 X22,19,5,20 X20,13,21,14 X14,21,15,22 X2536 X4,12,1,11
Gauss code {1, -10, 2, -11}, {10, -1, 3, -6, 5, -2, 11, -3, 8, -9, 4, -5, 6, -4, 7, -8, 9, -7}
A Braid Representative
A Morse Link Presentation L11a20 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1)^5}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial -q^{7/2}+4 q^{5/2}-9 q^{3/2}+14 \sqrt{q}-\frac{19}{\sqrt{q}}+\frac{20}{q^{3/2}}-\frac{21}{q^{5/2}}+\frac{17}{q^{7/2}}-\frac{12}{q^{9/2}}+\frac{7}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -z a^7-a^7 z^{-1} +3 z^3 a^5+6 z a^5+3 a^5 z^{-1} -3 z^5 a^3-9 z^3 a^3-10 z a^3-3 a^3 z^{-1} +z^7 a+4 z^5 a+8 z^3 a+7 z a+2 a z^{-1} -z^5 a^{-1} -2 z^3 a^{-1} -2 z a^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial -a^4 z^{10}-a^2 z^{10}-3 a^5 z^9-8 a^3 z^9-5 a z^9-4 a^6 z^8-12 a^4 z^8-17 a^2 z^8-9 z^8-3 a^7 z^7-4 a^5 z^7+a^3 z^7-6 a z^7-8 z^7 a^{-1} -a^8 z^6+6 a^6 z^6+30 a^4 z^6+40 a^2 z^6-4 z^6 a^{-2} +13 z^6+8 a^7 z^5+24 a^5 z^5+32 a^3 z^5+30 a z^5+13 z^5 a^{-1} -z^5 a^{-3} +3 a^8 z^4+2 a^6 z^4-22 a^4 z^4-32 a^2 z^4+5 z^4 a^{-2} -6 z^4-7 a^7 z^3-27 a^5 z^3-44 a^3 z^3-33 a z^3-8 z^3 a^{-1} +z^3 a^{-3} -3 a^8 z^2-5 a^6 z^2+5 a^4 z^2+10 a^2 z^2-z^2 a^{-2} +2 z^2+3 a^7 z+14 a^5 z+22 a^3 z+15 a z+4 z a^{-1} +a^8+2 a^6-2 a^2-a^7 z^{-1} -3 a^5 z^{-1} -3 a^3 z^{-1} -2 a z^{-1} - 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 \
8           11
6          3 -3
4         61 5
2        83  -5
0       116   5
-2      1110    -1
-4     109     1
-6    711      4
-8   510       -5
-10  27        5
-12 15         -4
-14 2          2
-161           -1
Integral Khovanov Homology

(db, data source)

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