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

Visit L11a338 at Knotilus!

Link Presentations

[edit Notes on L11a338's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X14,3,15,4 X20,13,21,14 X12,21,13,22 X22,5,9,6 X16,8,17,7 X18,16,19,15 X8,18,1,17 X6,9,7,10 X4,19,5,20
Gauss code {1, -2, 3, -11, 6, -10, 7, -9}, {10, -1, 2, -5, 4, -3, 8, -7, 9, -8, 11, -4, 5, -6}
A Braid Representative
A Morse Link Presentation L11a338 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 u^3 v^3-3 u^3 v^2+u^3 v-4 u^2 v^3+12 u^2 v^2-9 u^2 v+2 u^2+2 u v^3-9 u v^2+12 u v-4 u+v^2-3 v+2}{u^{3/2} v^{3/2}} (db)
Jones polynomial \frac{21}{q^{9/2}}-\frac{22}{q^{7/2}}+\frac{18}{q^{5/2}}+q^{3/2}-\frac{15}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{3}{q^{17/2}}-\frac{7}{q^{15/2}}+\frac{13}{q^{13/2}}-\frac{18}{q^{11/2}}-4 \sqrt{q}+\frac{9}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial z^5 a^7+3 z^3 a^7+3 z a^7-z^7 a^5-4 z^5 a^5-7 z^3 a^5-4 z a^5+a^5 z^{-1} -z^7 a^3-3 z^5 a^3-3 z^3 a^3-2 z a^3-a^3 z^{-1} +z^5 a+2 z^3 a+z a (db)
Kauffman polynomial a^{11} z^5-2 a^{11} z^3+a^{11} z+3 a^{10} z^6-5 a^{10} z^4+2 a^{10} z^2+5 a^9 z^7-6 a^9 z^5+a^9 z^3+7 a^8 z^8-12 a^8 z^6+13 a^8 z^4-7 a^8 z^2+6 a^7 z^9-7 a^7 z^7+4 a^7 z^5+a^7 z+2 a^6 z^{10}+12 a^6 z^8-38 a^6 z^6+42 a^6 z^4-15 a^6 z^2+12 a^5 z^9-22 a^5 z^7+9 a^5 z^5+2 a^5 z^3+a^5 z^{-1} +2 a^4 z^{10}+12 a^4 z^8-39 a^4 z^6+33 a^4 z^4-9 a^4 z^2-a^4+6 a^3 z^9-6 a^3 z^7-11 a^3 z^5+11 a^3 z^3-4 a^3 z+a^3 z^{-1} +7 a^2 z^8-15 a^2 z^6+7 a^2 z^4-2 a^2 z^2+4 a z^7-9 a z^5+6 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        93  6
-4       107   -3
-6      128    4
-8     910     1
-10    912      -3
-12   510       5
-14  28        -6
-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}^{2}
r=-5 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-4 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r=-3 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=-1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
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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