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

Visit L11a531 at Knotilus!

Link Presentations

[edit Notes on L11a531's Link Presentations]

Planar diagram presentation X6172 X2536 X18,11,19,12 X10,3,11,4 X4,9,1,10 X14,7,15,8 X8,13,5,14 X22,19,13,20 X20,15,21,16 X16,21,17,22 X12,17,9,18
Gauss code {1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 3, -11}, {7, -6, 9, -10, 11, -3, 8, -9, 10, -8}
A Braid Representative
A Morse Link Presentation L11a531 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-2 u v w x^2+2 u v w x-u v w-u v x^3+3 u v x^2-3 u v x+u v-u w x^3+3 u w x^2-3 u w x+u w+2 u x^3-4 u x^2+4 u x-u-v w x^3+4 v w x^2-4 v w x+2 v w+v x^3-3 v x^2+3 v x-v+w x^3-3 w x^2+3 w x-w-x^3+2 x^2-2 x}{\sqrt{u} \sqrt{v} \sqrt{w} x^{3/2}} (db)
Jones polynomial -\frac{11}{q^{9/2}}+\frac{4}{q^{7/2}}-\frac{1}{q^{5/2}}+\frac{1}{q^{27/2}}-\frac{2}{q^{25/2}}+\frac{6}{q^{23/2}}-\frac{11}{q^{21/2}}+\frac{16}{q^{19/2}}-\frac{22}{q^{17/2}}+\frac{19}{q^{15/2}}-\frac{21}{q^{13/2}}+\frac{14}{q^{11/2}} (db)
Signature -5 (db)
HOMFLY-PT polynomial -a^{15} z^{-3} +5 a^{13} z^{-3} +5 a^{13} z^{-1} -9 a^{11} z^{-3} -10 a^{11} z-19 a^{11} z^{-1} +10 a^9 z^3+7 a^9 z^{-3} +26 a^9 z+23 a^9 z^{-1} -4 a^7 z^5-13 a^7 z^3-2 a^7 z^{-3} -16 a^7 z-9 a^7 z^{-1} -a^5 z^5-a^5 z^3 (db)
Kauffman polynomial -z^6 a^{16}+4 z^4 a^{16}-6 z^2 a^{16}-a^{16} z^{-2} +4 a^{16}-2 z^7 a^{15}+5 z^5 a^{15}-4 z^3 a^{15}+2 z a^{15}-2 a^{15} z^{-1} +a^{15} z^{-3} -2 z^8 a^{14}-3 z^6 a^{14}+22 z^4 a^{14}-34 z^2 a^{14}-7 a^{14} z^{-2} +24 a^{14}-2 z^9 a^{13}-5 z^7 a^{13}+19 z^5 a^{13}-22 z^3 a^{13}+16 z a^{13}-12 a^{13} z^{-1} +5 a^{13} z^{-3} -z^{10} a^{12}-7 z^8 a^{12}+3 z^6 a^{12}+38 z^4 a^{12}-75 z^2 a^{12}-18 a^{12} z^{-2} +58 a^{12}-7 z^9 a^{11}-3 z^7 a^{11}+34 z^5 a^{11}-45 z^3 a^{11}+37 z a^{11}-24 a^{11} z^{-1} +9 a^{11} z^{-3} -z^{10} a^{10}-15 z^8 a^{10}+19 z^6 a^{10}+29 z^4 a^{10}-73 z^2 a^{10}-19 a^{10} z^{-2} +60 a^{10}-5 z^9 a^9-10 z^7 a^9+40 z^5 a^9-47 z^3 a^9+39 z a^9-23 a^9 z^{-1} +7 a^9 z^{-3} -10 z^8 a^8+10 z^6 a^8+12 z^4 a^8-26 z^2 a^8-7 a^8 z^{-2} +23 a^8-10 z^7 a^7+19 z^5 a^7-19 z^3 a^7+16 z a^7-9 a^7 z^{-1} +2 a^7 z^{-3} -4 z^6 a^6+3 z^4 a^6-z^5 a^5+z^3 a^5 (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           11
-6          41-3
-8         7  7
-10        74  -3
-12       147   7
-14      1113    2
-16     118     3
-18    511      6
-20   611       -5
-22  16        5
-24 15         -4
-26 1          1
-281           -1
Integral Khovanov Homology

(db, data source)

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