L11a73

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L11a72.gif

L11a72

L11a74.gif

L11a74

Contents

L11a73.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a73 at Knotilus!


Link Presentations

[edit Notes on L11a73's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X20,13,21,14 X16,7,17,8 X8,19,9,20 X18,9,19,10 X10,17,11,18 X22,15,5,16 X14,21,15,22 X2536 X4,11,1,12
Gauss code {1, -10, 2, -11}, {10, -1, 4, -5, 6, -7, 11, -2, 3, -9, 8, -4, 7, -6, 5, -3, 9, -8}
A Braid Representative
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A Morse Link Presentation L11a73 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 t(2)^5+4 t(1) t(2)^4-5 t(2)^4-7 t(1) t(2)^3+7 t(2)^3+7 t(1) t(2)^2-7 t(2)^2-5 t(1) t(2)+4 t(2)+2 t(1)}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial -\frac{1}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{10}{q^{11/2}}-\frac{15}{q^{13/2}}+\frac{16}{q^{15/2}}-\frac{16}{q^{17/2}}+\frac{13}{q^{19/2}}-\frac{10}{q^{21/2}}+\frac{6}{q^{23/2}}-\frac{2}{q^{25/2}}+\frac{1}{q^{27/2}} (db)
Signature -5 (db)
HOMFLY-PT polynomial a^{13} (-z)-2 a^{13} z^{-1} +3 a^{11} z^3+7 a^{11} z+4 a^{11} z^{-1} -2 a^9 z^5-4 a^9 z^3-a^9 z-a^9 z^{-1} -3 a^7 z^5-9 a^7 z^3-7 a^7 z-a^7 z^{-1} -a^5 z^5-2 a^5 z^3 (db)
Kauffman polynomial a^{16} z^6-4 a^{16} z^4+5 a^{16} z^2-2 a^{16}+2 a^{15} z^7-5 a^{15} z^5+2 a^{15} z^3+a^{15} z+3 a^{14} z^8-6 a^{14} z^6+a^{14} z^4+2 a^{14} z^2-a^{14}+3 a^{13} z^9-6 a^{13} z^7+9 a^{13} z^5-16 a^{13} z^3+9 a^{13} z-2 a^{13} z^{-1} +a^{12} z^{10}+6 a^{12} z^8-22 a^{12} z^6+32 a^{12} z^4-25 a^{12} z^2+6 a^{12}+7 a^{11} z^9-18 a^{11} z^7+29 a^{11} z^5-31 a^{11} z^3+17 a^{11} z-4 a^{11} z^{-1} +a^{10} z^{10}+9 a^{10} z^8-30 a^{10} z^6+41 a^{10} z^4-22 a^{10} z^2+5 a^{10}+4 a^9 z^9-4 a^9 z^7-a^9 z^5+4 a^9 z^3+2 a^9 z-a^9 z^{-1} +6 a^8 z^8-12 a^8 z^6+9 a^8 z^4-a^8+6 a^7 z^7-15 a^7 z^5+15 a^7 z^3-7 a^7 z+a^7 z^{-1} +3 a^6 z^6-5 a^6 z^4+a^5 z^5-2 a^5 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 \
-11-10-9-8-7-6-5-4-3-2-10χ
-4           11
-6          31-2
-8         4  4
-10        63  -3
-12       94   5
-14      87    -1
-16     88     0
-18    58      3
-20   58       -3
-22  15        4
-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}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-7 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-6 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-5 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}_2^{3} {\mathbb Z}^{3}
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

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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L11a72

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L11a74