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

Visit L11a352 at Knotilus!

Link Presentations

[edit Notes on L11a352's Link Presentations]

Planar diagram presentation X12,1,13,2 X8493 X14,5,15,6 X20,7,21,8 X16,13,17,14 X18,10,19,9 X10,11,1,12 X6,15,7,16 X4,21,5,22 X22,17,11,18 X2,20,3,19
Gauss code {1, -11, 2, -9, 3, -8, 4, -2, 6, -7}, {7, -1, 5, -3, 8, -5, 10, -6, 11, -4, 9, -10}
A Braid Representative
A Morse Link Presentation L11a352 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 u^3 v^3-4 u^3 v^2+3 u^3 v-u^3-5 u^2 v^3+13 u^2 v^2-11 u^2 v+4 u^2+4 u v^3-11 u v^2+13 u v-5 u-v^3+3 v^2-4 v+2}{u^{3/2} v^{3/2}} (db)
Jones polynomial \frac{28}{q^{9/2}}-\frac{28}{q^{7/2}}+\frac{23}{q^{5/2}}+q^{3/2}-\frac{17}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{5}{q^{17/2}}-\frac{12}{q^{15/2}}+\frac{19}{q^{13/2}}-\frac{25}{q^{11/2}}-4 \sqrt{q}+\frac{9}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial z^5 a^7+z^3 a^7-z^7 a^5-2 z^5 a^5+2 z a^5+a^5 z^{-1} -z^7 a^3-3 z^5 a^3-5 z^3 a^3-5 z a^3-a^3 z^{-1} +z^5 a+2 z^3 a+z a (db)
Kauffman polynomial a^{11} z^5+5 a^{10} z^6-3 a^{10} z^4+12 a^9 z^7-15 a^9 z^5+5 a^9 z^3-a^9 z+16 a^8 z^8-24 a^8 z^6+11 a^8 z^4-3 a^8 z^2+11 a^7 z^9-3 a^7 z^7-20 a^7 z^5+13 a^7 z^3-2 a^7 z+3 a^6 z^{10}+24 a^6 z^8-58 a^6 z^6+36 a^6 z^4-6 a^6 z^2+18 a^5 z^9-24 a^5 z^7-8 a^5 z^5+19 a^5 z^3-7 a^5 z+a^5 z^{-1} +3 a^4 z^{10}+15 a^4 z^8-43 a^4 z^6+31 a^4 z^4-5 a^4 z^2-a^4+7 a^3 z^9-5 a^3 z^7-13 a^3 z^5+18 a^3 z^3-8 a^3 z+a^3 z^{-1} +7 a^2 z^8-13 a^2 z^6+7 a^2 z^4-a^2 z^2+4 a z^7-9 a z^5+7 a z^3-2 a z+z^6-2 z^4+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 \
4           1-1
2          3 3
0         61 -5
-2        113  8
-4       137   -6
-6      1510    5
-8     1313     0
-10    1215      -3
-12   814       6
-14  411        -7
-16 18         7
-18 4          -4
-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}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-5 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-4 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{12}
r=-3 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=-2 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{15} {\mathbb Z}^{15}
r=-1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{11}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
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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