L11n374

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L11n373.gif

L11n373

L11n375.gif

L11n375

Contents

L11n374.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n374 at Knotilus!


Link Presentations

[edit Notes on L11n374's Link Presentations]

Planar diagram presentation X6172 X2,9,3,10 X12,3,13,4 X10,5,11,6 X16,11,5,12 X4,15,1,16 X13,20,14,21 X7,19,8,18 X17,9,18,8 X19,22,20,17 X21,14,22,15
Gauss code {1, -2, 3, -6}, {-9, 8, -10, 7, -11, 10}, {4, -1, -8, 9, 2, -4, 5, -3, -7, 11, 6, -5}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart4.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n374 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(v-1) (w-1) \left(2 u w^2-u w+w-2\right)}{\sqrt{u} \sqrt{v} w^{3/2}} (db)
Jones polynomial - q^{-9} +3 q^{-8} -5 q^{-7} +8 q^{-6} -8 q^{-5} +8 q^{-4} -6 q^{-3} +6 q^{-2} -2 q^{-1} +1 (db)
Signature -4 (db)
HOMFLY-PT polynomial -a^{10}+z^4 a^8+4 z^2 a^8+3 a^8-z^6 a^6-4 z^4 a^6-5 z^2 a^6+a^6 z^{-2} -2 a^6-z^6 a^4-3 z^4 a^4-2 z^2 a^4-2 a^4 z^{-2} -3 a^4+z^4 a^2+3 z^2 a^2+a^2 z^{-2} +3 a^2 (db)
Kauffman polynomial z^3 a^{11}-2 z a^{11}+3 z^4 a^{10}-4 z^2 a^{10}+a^{10}+2 z^7 a^9-7 z^5 a^9+14 z^3 a^9-6 z a^9+3 z^8 a^8-12 z^6 a^8+22 z^4 a^8-12 z^2 a^8+3 a^8+z^9 a^7+2 z^7 a^7-14 z^5 a^7+20 z^3 a^7-6 z a^7+5 z^8 a^6-15 z^6 a^6+14 z^4 a^6-5 z^2 a^6+a^6 z^{-2} -a^6+z^9 a^5+2 z^7 a^5-12 z^5 a^5+7 z^3 a^5+2 z a^5-2 a^5 z^{-1} +2 z^8 a^4-2 z^6 a^4-9 z^4 a^4+9 z^2 a^4+2 a^4 z^{-2} -6 a^4+2 z^7 a^3-5 z^5 a^3+4 z a^3-2 a^3 z^{-1} +z^6 a^2-4 z^4 a^2+6 z^2 a^2+a^2 z^{-2} -4 a^2 (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 \
-7-6-5-4-3-2-1012χ
1         11
-1        1 -1
-3       51 4
-5      33  0
-7     53   2
-9   143    0
-11   55     0
-13  25      3
-15 13       -2
-17 2        2
-191         -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-5 i=-3 i=-1
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{5} {\mathbb Z}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{5}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
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).

See/edit the Link_Splice_Base (expert).

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L11n373

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L11n375