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

Visit L11a312 at Knotilus!

Link Presentations

[edit Notes on L11a312's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X8,9,1,10 X22,13,9,14 X14,8,15,7 X18,6,19,5 X20,17,21,18 X16,21,17,22 X6,16,7,15 X4,20,5,19
Gauss code {1, -2, 3, -11, 7, -10, 6, -4}, {4, -1, 2, -3, 5, -6, 10, -9, 8, -7, 11, -8, 9, -5}
A Braid Representative
A Morse Link Presentation L11a312 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1)-1) (t(2)-1) \left(t(2)^2 t(1)^2-2 t(2) t(1)^2-2 t(2)^2 t(1)-2 t(1)-2 t(2)+1\right)}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial -q^{5/2}+2 q^{3/2}-5 \sqrt{q}+\frac{7}{\sqrt{q}}-\frac{10}{q^{3/2}}+\frac{12}{q^{5/2}}-\frac{13}{q^{7/2}}+\frac{11}{q^{9/2}}-\frac{9}{q^{11/2}}+\frac{6}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{1}{q^{17/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial -z^3 a^7-2 z a^7+2 z^5 a^5+6 z^3 a^5+3 z a^5-z^7 a^3-4 z^5 a^3-5 z^3 a^3-3 z a^3+2 z^5 a+7 z^3 a+5 z a+a z^{-1} -z^3 a^{-1} -3 z a^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial a^{10} z^4-a^{10} z^2+3 a^9 z^5-3 a^9 z^3+5 a^8 z^6-6 a^8 z^4+2 a^8 z^2+6 a^7 z^7-10 a^7 z^5+9 a^7 z^3-3 a^7 z+5 a^6 z^8-8 a^6 z^6+7 a^6 z^4-3 a^6 z^2+3 a^5 z^9-2 a^5 z^7-5 a^5 z^5+6 a^5 z^3-2 a^5 z+a^4 z^{10}+4 a^4 z^8-16 a^4 z^6+15 a^4 z^4-5 a^4 z^2+5 a^3 z^9-14 a^3 z^7+10 a^3 z^5-3 a^3 z^3+a^2 z^{10}+a^2 z^8-11 a^2 z^6+9 a^2 z^4+2 a z^9-5 a z^7+z^7 a^{-1} -3 a z^5-5 z^5 a^{-1} +11 a z^3+8 z^3 a^{-1} -6 a z+a z^{-1} -5 z a^{-1} + a^{-1} z^{-1} +2 z^8-8 z^6+8 z^4-z^2-1 (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 \
6           11
4          1 -1
2         41 3
0        31  -2
-2       74   3
-4      75    -2
-6     65     1
-8    57      2
-10   46       -2
-12  25        3
-14 14         -3
-16 2          2
-181           -1
Integral Khovanov Homology

(db, data source)

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