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

Visit L11a534 at Knotilus!

Link Presentations

[edit Notes on L11a534's Link Presentations]

Planar diagram presentation X6172 X2536 X18,11,19,12 X10,3,11,4 X4,9,1,10 X16,7,17,8 X8,15,5,16 X20,14,21,13 X22,19,15,20 X12,22,13,21 X14,17,9,18
Gauss code {1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 3, -10, 8, -11}, {7, -6, 11, -3, 9, -8, 10, -9}
A Braid Representative
A Morse Link Presentation L11a534 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1) t(4)^2 t(3)^2+t(2) t(4)^2 t(3)^2-t(4)^2 t(3)^2+t(1) t(3)^2-t(1) t(2) t(3)^2+2 t(2) t(3)^2-2 t(1) t(4) t(3)^2+t(1) t(2) t(4) t(3)^2-3 t(2) t(4) t(3)^2+2 t(4) t(3)^2-t(3)^2-3 t(1) t(4)^2 t(3)+t(1) t(2) t(4)^2 t(3)-2 t(2) t(4)^2 t(3)+2 t(4)^2 t(3)-2 t(1) t(3)+2 t(1) t(2) t(3)-3 t(2) t(3)+6 t(1) t(4) t(3)-4 t(1) t(2) t(4) t(3)+6 t(2) t(4) t(3)-4 t(4) t(3)+t(3)+2 t(1) t(4)^2-t(1) t(2) t(4)^2+t(2) t(4)^2-t(4)^2+t(1)-t(1) t(2)+t(2)-3 t(1) t(4)+2 t(1) t(2) t(4)-2 t(2) t(4)+t(4)}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)} (db)
Jones polynomial \frac{21}{q^{9/2}}-\frac{21}{q^{7/2}}+\frac{14}{q^{5/2}}-\frac{10}{q^{3/2}}+\frac{1}{q^{21/2}}-\frac{3}{q^{19/2}}+\frac{7}{q^{17/2}}-\frac{14}{q^{15/2}}+\frac{17}{q^{13/2}}-\frac{23}{q^{11/2}}-\sqrt{q}+\frac{4}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial -a^{11} z^{-1} +4 z a^9+4 a^9 z^{-1} +a^9 z^{-3} -6 z^3 a^7-11 z a^7-9 a^7 z^{-1} -3 a^7 z^{-3} +3 z^5 a^5+8 z^3 a^5+13 z a^5+10 a^5 z^{-1} +3 a^5 z^{-3} +z^5 a^3-2 z^3 a^3-6 z a^3-4 a^3 z^{-1} -a^3 z^{-3} -z^3 a (db)
Kauffman polynomial a^{12} z^6-3 a^{12} z^4+3 a^{12} z^2-a^{12}+3 a^{11} z^7-8 a^{11} z^5+9 a^{11} z^3-6 a^{11} z+2 a^{11} z^{-1} +4 a^{10} z^8-4 a^{10} z^6-8 a^{10} z^4+14 a^{10} z^2-6 a^{10}+3 a^9 z^9+7 a^9 z^7-33 a^9 z^5+42 a^9 z^3-a^9 z^{-3} -30 a^9 z+11 a^9 z^{-1} +a^8 z^{10}+13 a^8 z^8-24 a^8 z^6-4 a^8 z^4+28 a^8 z^2+3 a^8 z^{-2} -18 a^8+8 a^7 z^9+7 a^7 z^7-56 a^7 z^5+74 a^7 z^3-3 a^7 z^{-3} -49 a^7 z+18 a^7 z^{-1} +a^6 z^{10}+19 a^6 z^8-37 a^6 z^6+9 a^6 z^4+24 a^6 z^2+6 a^6 z^{-2} -21 a^6+5 a^5 z^9+12 a^5 z^7-48 a^5 z^5+56 a^5 z^3-3 a^5 z^{-3} -35 a^5 z+14 a^5 z^{-1} +10 a^4 z^8-14 a^4 z^6+4 a^4 z^4+7 a^4 z^2+3 a^4 z^{-2} -9 a^4+9 a^3 z^7-16 a^3 z^5+14 a^3 z^3-a^3 z^{-3} -10 a^3 z+5 a^3 z^{-1} +4 a^2 z^6-4 a^2 z^4+a z^5-a z^3 (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 \
2           11
0          3 -3
-2         71 6
-4        84  -4
-6       136   7
-8      1111    0
-10     1210     2
-12    814      6
-14   69       -3
-16  29        7
-18 15         -4
-20 2          2
-221           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-4 i=-2
r=-9 {\mathbb Z}
r=-8 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-6 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=-5 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-4 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{12}
r=-3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=-2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{13}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\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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