L11a62

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L11a61

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L11a63

Contents

L11a62.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11a62's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X22,13,5,14 X14,7,15,8 X20,18,21,17 X10,15,11,16 X8,19,9,20 X18,9,19,10 X16,22,17,21 X2536 X4,11,1,12
Gauss code {1, -10, 2, -11}, {10, -1, 4, -7, 8, -6, 11, -2, 3, -4, 6, -9, 5, -8, 7, -5, 9, -3}
A Braid Representative
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A Morse Link Presentation L11a62 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(2)^5+2 t(1) t(2)^4-6 t(2)^4-10 t(1) t(2)^3+12 t(2)^3+12 t(1) t(2)^2-10 t(2)^2-6 t(1) t(2)+2 t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial -\sqrt{q}+\frac{4}{\sqrt{q}}-\frac{9}{q^{3/2}}+\frac{13}{q^{5/2}}-\frac{18}{q^{7/2}}+\frac{20}{q^{9/2}}-\frac{20}{q^{11/2}}+\frac{16}{q^{13/2}}-\frac{12}{q^{15/2}}+\frac{7}{q^{17/2}}-\frac{3}{q^{19/2}}+\frac{1}{q^{21/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial -a^{11} z^{-1} +4 a^9 z+3 a^9 z^{-1} -5 a^7 z^3-7 a^7 z-3 a^7 z^{-1} +2 a^5 z^5+3 a^5 z^3+4 a^5 z+2 a^5 z^{-1} +a^3 z^5-a^3 z^3-3 a^3 z-a^3 z^{-1} -a z^3 (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+7 a^{11} z^3-3 a^{11} z+a^{11} z^{-1} +4 a^{10} z^8-6 a^{10} z^6-3 a^{10} z^4+6 a^{10} z^2-2 a^{10}+3 a^9 z^9+3 a^9 z^7-21 a^9 z^5+23 a^9 z^3-13 a^9 z+3 a^9 z^{-1} +a^8 z^{10}+11 a^8 z^8-28 a^8 z^6+21 a^8 z^4-6 a^8 z^2+8 a^7 z^9-5 a^7 z^7-20 a^7 z^5+31 a^7 z^3-16 a^7 z+3 a^7 z^{-1} +a^6 z^{10}+16 a^6 z^8-41 a^6 z^6+38 a^6 z^4-13 a^6 z^2+2 a^6+5 a^5 z^9+3 a^5 z^7-22 a^5 z^5+24 a^5 z^3-10 a^5 z+2 a^5 z^{-1} +9 a^4 z^8-16 a^4 z^6+12 a^4 z^4-4 a^4 z^2+8 a^3 z^7-14 a^3 z^5+8 a^3 z^3-4 a^3 z+a^3 z^{-1} +4 a^2 z^6-5 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 \
-9-8-7-6-5-4-3-2-1012χ
2           11
0          3 -3
-2         61 5
-4        84  -4
-6       105   5
-8      108    -2
-10     1010     0
-12    711      4
-14   59       -4
-16  27        5
-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}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-5 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-4 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=-3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
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

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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L11a61

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L11a63