L11a507

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L11a506.gif

L11a506

L11a508.gif

L11a508

Contents

L11a507.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a507 at Knotilus!


Link Presentations

[edit Notes on L11a507's Link Presentations]

Planar diagram presentation X8192 X18,12,19,11 X10,4,11,3 X2,20,3,19 X16,8,17,7 X20,9,21,10 X12,18,7,17 X22,16,13,15 X14,6,15,5 X4,14,5,13 X6,21,1,22
Gauss code {1, -4, 3, -10, 9, -11}, {5, -1, 6, -3, 2, -7}, {10, -9, 8, -5, 7, -2, 4, -6, 11, -8}
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.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gif
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A Morse Link Presentation L11a507 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(w-1) \left(u^2 v^2 w+u^2 \left(-v^2\right)+u^2 v w^2-2 u^2 v w+u^2 v-u^2 w^2+u^2 w-2 u v^2 w+2 u v^2-2 u v w^2+5 u v w-2 u v+2 u w^2-2 u w+v^2 w-v^2+v w^2-2 v w+v-w^2+w\right)}{u v w^{3/2}} (db)
Jones polynomial q^9-4 q^8+8 q^7-13 q^6+19 q^5-20 q^4+22 q^3-18 q^2- q^{-2} +14 q+4 q^{-1} -8 (db)
Signature 2 (db)
HOMFLY-PT polynomial z^2 a^{-8} -2 z^4 a^{-6} -2 z^2 a^{-6} + a^{-6} z^{-2} +z^6 a^{-4} +z^4 a^{-4} -2 a^{-4} z^{-2} -2 a^{-4} +z^6 a^{-2} +2 z^4 a^{-2} +3 z^2 a^{-2} + a^{-2} z^{-2} +2 a^{-2} -z^4-z^2 (db)
Kauffman polynomial 2 z^{10} a^{-4} +2 z^{10} a^{-6} +6 z^9 a^{-3} +12 z^9 a^{-5} +6 z^9 a^{-7} +8 z^8 a^{-2} +12 z^8 a^{-4} +11 z^8 a^{-6} +7 z^8 a^{-8} +7 z^7 a^{-1} -3 z^7 a^{-3} -24 z^7 a^{-5} -10 z^7 a^{-7} +4 z^7 a^{-9} -10 z^6 a^{-2} -35 z^6 a^{-4} -41 z^6 a^{-6} -19 z^6 a^{-8} +z^6 a^{-10} +4 z^6+a z^5-10 z^5 a^{-1} -6 z^5 a^{-3} +14 z^5 a^{-5} -z^5 a^{-7} -10 z^5 a^{-9} +2 z^4 a^{-2} +34 z^4 a^{-4} +44 z^4 a^{-6} +16 z^4 a^{-8} -2 z^4 a^{-10} -6 z^4-a z^3+3 z^3 a^{-1} +4 z^3 a^{-3} +5 z^3 a^{-7} +5 z^3 a^{-9} +z^2 a^{-2} -10 z^2 a^{-4} -15 z^2 a^{-6} -6 z^2 a^{-8} +2 z^2+2 z a^{-3} +2 z a^{-5} -2 a^{-2} -3 a^{-4} -2 a^{-6} -2 a^{-3} z^{-1} -2 a^{-5} z^{-1} + a^{-2} z^{-2} +2 a^{-4} z^{-2} + a^{-6} 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 \
-3-2-1012345678χ
19           11
17          3 -3
15         51 4
13        83  -5
11       115   6
9      109    -1
7     1210     2
5    812      4
3   610       -4
1  39        6
-1 15         -4
-3 3          3
-51           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{12}
r=3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=4 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{11}
r=5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=8 {\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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L11a506

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L11a508