L11n410

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L11n409.gif

L11n409

L11n411.gif

L11n411

Contents

L11n410.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n410 at Knotilus!


Link Presentations

[edit Notes on L11n410's Link Presentations]

Planar diagram presentation X6172 X3,13,4,12 X13,22,14,19 X7,20,8,21 X19,10,20,11 X9,16,10,17 X17,14,18,15 X15,8,16,9 X21,18,22,5 X2536 X11,1,12,4
Gauss code {1, -10, -2, 11}, {-5, 4, -9, 3}, {10, -1, -4, 8, -6, 5, -11, 2, -3, 7, -8, 6, -7, 9}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n410 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u-1) (w-1) \left(2 v w^2-2 v w+v-w^2+2 w-2\right)}{\sqrt{u} \sqrt{v} w^{3/2}} (db)
Jones polynomial 1-3 q^{-1} +7 q^{-2} -9 q^{-3} +14 q^{-4} -13 q^{-5} +13 q^{-6} -10 q^{-7} +7 q^{-8} -3 q^{-9} (db)
Signature -4 (db)
HOMFLY-PT polynomial -a^{10} z^{-2} -a^{10}+a^8 z^4+3 a^8 z^2+4 a^8 z^{-2} +6 a^8-a^6 z^6-3 a^6 z^4-6 a^6 z^2-5 a^6 z^{-2} -10 a^6-a^4 z^6-2 a^4 z^4+a^4 z^2+2 a^4 z^{-2} +4 a^4+a^2 z^4+2 a^2 z^2+a^2 (db)
Kauffman polynomial 6 z^3 a^{11}-5 z a^{11}+a^{11} z^{-1} +3 z^6 a^{10}+3 z^4 a^{10}-6 z^2 a^{10}-a^{10} z^{-2} +3 a^{10}+8 z^7 a^9-19 z^5 a^9+29 z^3 a^9-20 z a^9+5 a^9 z^{-1} +7 z^8 a^8-16 z^6 a^8+25 z^4 a^8-23 z^2 a^8-4 a^8 z^{-2} +13 a^8+2 z^9 a^7+9 z^7 a^7-33 z^5 a^7+41 z^3 a^7-30 z a^7+9 a^7 z^{-1} +11 z^8 a^6-29 z^6 a^6+30 z^4 a^6-25 z^2 a^6-5 a^6 z^{-2} +16 a^6+2 z^9 a^5+4 z^7 a^5-22 z^5 a^5+23 z^3 a^5-15 z a^5+5 a^5 z^{-1} +4 z^8 a^4-9 z^6 a^4+5 z^4 a^4-5 z^2 a^4-2 a^4 z^{-2} +6 a^4+3 z^7 a^3-8 z^5 a^3+5 z^3 a^3+z^6 a^2-3 z^4 a^2+3 z^2 a^2-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        2 -2
-3       51 4
-5      64  -2
-7     83   5
-9    56    1
-11   88     0
-13  47      3
-15 36       -3
-17 4        4
-193         -3
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-5 i=-3
r=-7 {\mathbb Z}^{3}
r=-6 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-5 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{8}
r=-3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{5}
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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L11n409

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L11n411