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

Visit L11a205 at Knotilus!

Link Presentations

[edit Notes on L11a205's Link Presentations]

Planar diagram presentation X8192 X2,9,3,10 X10,3,11,4 X6718 X18,15,19,16 X16,6,17,5 X4,18,5,17 X20,11,21,12 X22,13,7,14 X12,21,13,22 X14,19,15,20
Gauss code {1, -2, 3, -7, 6, -4}, {4, -1, 2, -3, 8, -10, 9, -11, 5, -6, 7, -5, 11, -8, 10, -9}
A Braid Representative
A Morse Link Presentation L11a205 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 t(1)^2 t(2)^4-2 t(1) t(2)^4-3 t(1)^2 t(2)^3+4 t(1) t(2)^3-2 t(2)^3+3 t(1)^2 t(2)^2-5 t(1) t(2)^2+3 t(2)^2-2 t(1)^2 t(2)+4 t(1) t(2)-3 t(2)-2 t(1)+2}{t(1) t(2)^2} (db)
Jones polynomial -\frac{9}{q^{9/2}}+\frac{6}{q^{7/2}}-\frac{4}{q^{5/2}}+\frac{1}{q^{3/2}}+\frac{1}{q^{23/2}}-\frac{3}{q^{21/2}}+\frac{6}{q^{19/2}}-\frac{9}{q^{17/2}}+\frac{11}{q^{15/2}}-\frac{12}{q^{13/2}}+\frac{11}{q^{11/2}}-\frac{1}{\sqrt{q}} (db)
Signature -5 (db)
HOMFLY-PT polynomial -z^5 a^9-3 z^3 a^9-2 z a^9+z^7 a^7+4 z^5 a^7+4 z^3 a^7-a^7 z^{-1} +z^7 a^5+5 z^5 a^5+9 z^3 a^5+8 z a^5+3 a^5 z^{-1} -z^5 a^3-5 z^3 a^3-7 z a^3-2 a^3 z^{-1} (db)
Kauffman polynomial -z^4 a^{14}+z^2 a^{14}-3 z^5 a^{13}+3 z^3 a^{13}-5 z^6 a^{12}+6 z^4 a^{12}-2 z^2 a^{12}-6 z^7 a^{11}+10 z^5 a^{11}-8 z^3 a^{11}+z a^{11}-5 z^8 a^{10}+9 z^6 a^{10}-9 z^4 a^{10}+3 z^2 a^{10}-3 z^9 a^9+5 z^7 a^9-7 z^5 a^9+8 z^3 a^9-3 z a^9-z^{10} a^8-z^8 a^8+3 z^6 a^8+2 z^4 a^8-2 z^2 a^8+a^8-4 z^9 a^7+13 z^7 a^7-18 z^5 a^7+15 z^3 a^7-2 z a^7-a^7 z^{-1} -z^{10} a^6+3 z^8 a^6-8 z^6 a^6+19 z^4 a^6-15 z^2 a^6+3 a^6-z^9 a^5+z^7 a^5+8 z^5 a^5-16 z^3 a^5+11 z a^5-3 a^5 z^{-1} -z^8 a^4+3 z^6 a^4+z^4 a^4-7 z^2 a^4+3 a^4-z^7 a^3+6 z^5 a^3-12 z^3 a^3+9 z a^3-2 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         41 3
-6        31  -2
-8       63   3
-10      64    -2
-12     65     1
-14    56      1
-16   46       -2
-18  25        3
-20 14         -3
-22 2          2
-241           -1
Integral Khovanov Homology

(db, data source)

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

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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