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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 L11a211's Link Presentations]

Planar diagram presentation X8192 X2,9,3,10 X10,3,11,4 X16,5,17,6 X14,7,15,8 X22,15,7,16 X20,18,21,17 X18,12,19,11 X12,20,13,19 X4,13,5,14 X6,21,1,22
Gauss code {1, -2, 3, -10, 4, -11}, {5, -1, 2, -3, 8, -9, 10, -5, 6, -4, 7, -8, 9, -7, 11, -6}
A Braid Representative
A Morse Link Presentation L11a211 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 t(1)^2 t(2)^4-2 t(1) t(2)^4-5 t(1)^2 t(2)^3+8 t(1) t(2)^3-2 t(2)^3+6 t(1)^2 t(2)^2-11 t(1) t(2)^2+6 t(2)^2-2 t(1)^2 t(2)+8 t(1) t(2)-5 t(2)-2 t(1)+2}{t(1) t(2)^2} (db)
Jones polynomial q^{3/2}-4 \sqrt{q}+\frac{8}{\sqrt{q}}-\frac{13}{q^{3/2}}+\frac{17}{q^{5/2}}-\frac{20}{q^{7/2}}+\frac{19}{q^{9/2}}-\frac{17}{q^{11/2}}+\frac{12}{q^{13/2}}-\frac{7}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{1}{q^{19/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial z^5 a^7+3 z^3 a^7+3 z a^7+a^7 z^{-1} -z^7 a^5-4 z^5 a^5-7 z^3 a^5-6 z a^5-a^5 z^{-1} -z^7 a^3-3 z^5 a^3-2 z^3 a^3+z^5 a+2 z^3 a (db)
Kauffman polynomial -z^5 a^{11}+2 z^3 a^{11}-z a^{11}-3 z^6 a^{10}+5 z^4 a^{10}-2 z^2 a^{10}-5 z^7 a^9+7 z^5 a^9-3 z^3 a^9+z a^9-6 z^8 a^8+8 z^6 a^8-5 z^4 a^8+2 z^2 a^8-5 z^9 a^7+5 z^7 a^7-4 z^5 a^7+5 z^3 a^7-4 z a^7+a^7 z^{-1} -2 z^{10} a^6-7 z^8 a^6+21 z^6 a^6-22 z^4 a^6+9 z^2 a^6-a^6-11 z^9 a^5+26 z^7 a^5-27 z^5 a^5+16 z^3 a^5-6 z a^5+a^5 z^{-1} -2 z^{10} a^4-8 z^8 a^4+30 z^6 a^4-28 z^4 a^4+8 z^2 a^4-6 z^9 a^3+12 z^7 a^3-5 z^5 a^3+z^3 a^3-7 z^8 a^2+19 z^6 a^2-14 z^4 a^2+3 z^2 a^2-4 z^7 a+10 z^5 a-5 z^3 a-z^6+2 z^4 (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 \
4           1-1
2          3 3
0         51 -4
-2        83  5
-4       106   -4
-6      107    3
-8     910     1
-10    810      -2
-12   49       5
-14  38        -5
-16 15         4
-18 2          -2
-201           1
Integral Khovanov Homology

(db, data source)

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