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

Visit L11a128 at Knotilus!

Link Presentations

[edit Notes on L11a128's Link Presentations]

Planar diagram presentation X6172 X14,3,15,4 X16,10,17,9 X18,12,19,11 X10,18,11,17 X12,16,13,15 X22,19,5,20 X20,7,21,8 X8,21,9,22 X2536 X4,13,1,14
Gauss code {1, -10, 2, -11}, {10, -1, 8, -9, 3, -5, 4, -6, 11, -2, 6, -3, 5, -4, 7, -8, 9, -7}
A Braid Representative
A Morse Link Presentation L11a128 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{\left(v^2-v+1\right)^2 (u v-2 u-2 v+1)}{\sqrt{u} v^{5/2}} (db)
Jones polynomial -q^{7/2}+4 q^{5/2}-8 q^{3/2}+13 \sqrt{q}-\frac{16}{\sqrt{q}}+\frac{17}{q^{3/2}}-\frac{18}{q^{5/2}}+\frac{13}{q^{7/2}}-\frac{10}{q^{9/2}}+\frac{5}{q^{11/2}}-\frac{2}{q^{13/2}}+\frac{1}{q^{15/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -z a^7-2 a^7 z^{-1} +3 z^3 a^5+9 z a^5+7 a^5 z^{-1} -3 z^5 a^3-11 z^3 a^3-15 z a^3-7 a^3 z^{-1} +z^7 a+4 z^5 a+7 z^3 a+6 z a+2 a z^{-1} -z^5 a^{-1} -2 z^3 a^{-1} -z a^{-1} (db)
Kauffman polynomial -a^4 z^{10}-a^2 z^{10}-2 a^5 z^9-6 a^3 z^9-4 a z^9-2 a^6 z^8-4 a^4 z^8-9 a^2 z^8-7 z^8-2 a^7 z^7-a^5 z^7+6 a^3 z^7-2 a z^7-7 z^7 a^{-1} -a^8 z^6+5 a^4 z^6+16 a^2 z^6-4 z^6 a^{-2} +8 z^6+6 a^7 z^5+11 a^5 z^5+5 a^3 z^5+12 a z^5+11 z^5 a^{-1} -z^5 a^{-3} +4 a^8 z^4+13 a^6 z^4+11 a^4 z^4-2 a^2 z^4+6 z^4 a^{-2} +2 z^4-6 a^7 z^3-17 a^5 z^3-15 a^3 z^3-9 a z^3-4 z^3 a^{-1} +z^3 a^{-3} -5 a^8 z^2-18 a^6 z^2-24 a^4 z^2-13 a^2 z^2-2 z^2 a^{-2} -4 z^2+4 a^7 z+16 a^5 z+15 a^3 z+4 a z+z a^{-1} +2 a^8+8 a^6+13 a^4+8 a^2+2-2 a^7 z^{-1} -7 a^5 z^{-1} -7 a^3 z^{-1} -2 a 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 \
8           11
6          3 -3
4         51 4
2        83  -5
0       85   3
-2      109    -1
-4     87     1
-6    510      5
-8   58       -3
-10  16        5
-12 14         -3
-14 1          1
-161           -1
Integral Khovanov Homology

(db, data source)

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

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See/edit the Link Page master template (intermediate).

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