L11a92

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L11a91.gif

L11a91

L11a93.gif

L11a93

Contents

L11a92.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11a92's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X20,8,21,7 X22,17,5,18 X18,21,19,22 X16,13,17,14 X14,9,15,10 X10,15,11,16 X8,20,9,19 X2536 X4,11,1,12
Gauss code {1, -10, 2, -11}, {10, -1, 3, -9, 7, -8, 11, -2, 6, -7, 8, -6, 4, -5, 9, -3, 5, -4}
A Braid Representative
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A Morse Link Presentation L11a92 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(2)^5+3 t(1) t(2)^4-6 t(2)^4-10 t(1) t(2)^3+11 t(2)^3+11 t(1) t(2)^2-10 t(2)^2-6 t(1) t(2)+3 t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial -\sqrt{q}+\frac{3}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{13}{q^{5/2}}-\frac{18}{q^{7/2}}+\frac{20}{q^{9/2}}-\frac{20}{q^{11/2}}+\frac{17}{q^{13/2}}-\frac{13}{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 z a^9+4 a^9 z^{-1} -6 z^3 a^7-11 z a^7-6 a^7 z^{-1} +3 z^5 a^5+8 z^3 a^5+10 z a^5+5 a^5 z^{-1} +z^5 a^3-z^3 a^3-4 z a^3-2 a^3 z^{-1} -z^3 a-z 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+8 a^{11} z^3-4 a^{11} z+a^{11} z^{-1} +4 a^{10} z^8-5 a^{10} z^6-5 a^{10} z^4+9 a^{10} z^2-3 a^{10}+3 a^9 z^9+5 a^9 z^7-27 a^9 z^5+31 a^9 z^3-18 a^9 z+4 a^9 z^{-1} +a^8 z^{10}+11 a^8 z^8-22 a^8 z^6+4 a^8 z^4+7 a^8 z^2-3 a^8+7 a^7 z^9+4 a^7 z^7-41 a^7 z^5+52 a^7 z^3-29 a^7 z+6 a^7 z^{-1} +a^6 z^{10}+14 a^6 z^8-28 a^6 z^6+15 a^6 z^4-a^6+4 a^5 z^9+8 a^5 z^7-33 a^5 z^5+41 a^5 z^3-23 a^5 z+5 a^5 z^{-1} +7 a^4 z^8-9 a^4 z^6+5 a^4 z^4-a^4+6 a^3 z^7-10 a^3 z^5+10 a^3 z^3-7 a^3 z+2 a^3 z^{-1} +3 a^2 z^6-4 a^2 z^4+a^2 z^2+a z^5-2 a z^3+a z (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          2 -2
-2         61 5
-4        83  -5
-6       105   5
-8      108    -2
-10     1010     0
-12    811      3
-14   59       -4
-16  28        6
-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}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-5 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
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}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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L11a91

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L11a93