L11a417

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L11a416.gif

L11a416

L11a418.gif

L11a418

Contents

L11a417.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11a417's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X22,14,11,13 X20,16,21,15 X8,18,9,17 X16,8,17,7 X18,10,19,9 X10,20,5,19 X14,22,15,21 X2536 X4,11,1,12
Gauss code {1, -10, 2, -11}, {10, -1, 6, -5, 7, -8}, {11, -2, 3, -9, 4, -6, 5, -7, 8, -4, 9, -3}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a417 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 u v^2 w^2-2 u v^2 w-2 u v w^2+4 u v w-2 u v-2 u w+3 u-3 v^2 w^2+2 v^2 w+2 v w^2-4 v w+2 v+2 w-2}{\sqrt{u} v w} (db)
Jones polynomial -q^8+3 q^7-5 q^6+8 q^5-10 q^4+11 q^3+ q^{-3} -10 q^2- q^{-2} +9 q+4 q^{-1} -5 (db)
Signature 2 (db)
HOMFLY-PT polynomial z^6 a^{-2} +z^6 a^{-4} +3 z^4 a^{-2} +3 z^4 a^{-4} -z^4 a^{-6} -2 z^4+a^2 z^2+3 z^2 a^{-2} +2 z^2 a^{-4} -2 z^2 a^{-6} -7 z^2+3 a^2+3 a^{-2} -6+a^2 z^{-2} + a^{-2} z^{-2} -2 z^{-2} (db)
Kauffman polynomial z^{10} a^{-2} +z^{10} a^{-4} +z^9 a^{-1} +4 z^9 a^{-3} +3 z^9 a^{-5} -3 z^8 a^{-2} +4 z^8 a^{-6} +z^8+a z^7-z^7 a^{-1} -15 z^7 a^{-3} -9 z^7 a^{-5} +4 z^7 a^{-7} +a^2 z^6+8 z^6 a^{-2} -4 z^6 a^{-4} -10 z^6 a^{-6} +3 z^6 a^{-8} -2 a z^5+26 z^5 a^{-3} +14 z^5 a^{-5} -9 z^5 a^{-7} +z^5 a^{-9} -5 a^2 z^4-10 z^4 a^{-2} +10 z^4 a^{-4} +9 z^4 a^{-6} -7 z^4 a^{-8} -9 z^4-3 a z^3-5 z^3 a^{-1} -14 z^3 a^{-3} -6 z^3 a^{-5} +4 z^3 a^{-7} -2 z^3 a^{-9} +8 a^2 z^2+5 z^2 a^{-2} -4 z^2 a^{-4} -2 z^2 a^{-6} +2 z^2 a^{-8} +13 z^2+6 a z+6 z a^{-1} -5 a^2-3 a^{-2} + a^{-4} -8-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-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 \
-4-3-2-101234567χ
17           1-1
15          2 2
13         31 -2
11        52  3
9       53   -2
7      65    1
5     67     1
3    34      -1
1   37       4
-1  12        -1
-3  3         3
-511          0
-71           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-4 {\mathbb Z} {\mathbb Z}
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{6}
r=3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=7 {\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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L11a416

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L11a418