L11a93

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L11a92.gif

L11a92

L11a94.gif

L11a94

Contents

L11a93.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a93 at Knotilus!


Link Presentations

[edit Notes on L11a93's Link Presentations]

Planar diagram presentation X6172 X12,4,13,3 X20,8,21,7 X22,17,5,18 X18,21,19,22 X14,10,15,9 X16,12,17,11 X10,16,11,15 X8,20,9,19 X2536 X4,14,1,13
Gauss code {1, -10, 2, -11}, {10, -1, 3, -9, 6, -8, 7, -2, 11, -6, 8, -7, 4, -5, 9, -3, 5, -4}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation L11a93 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1) \left(v^4-4 v^3+3 v^2-4 v+1\right)}{\sqrt{u} v^{5/2}} (db)
Jones polynomial q^{17/2}-3 q^{15/2}+7 q^{13/2}-12 q^{11/2}+15 q^{9/2}-17 q^{7/2}+16 q^{5/2}-14 q^{3/2}+10 \sqrt{q}-\frac{6}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{1}{q^{5/2}} (db)
Signature 3 (db)
HOMFLY-PT polynomial z^3 a^{-7} +2 z a^{-7} + a^{-7} z^{-1} -2 z^5 a^{-5} -6 z^3 a^{-5} -7 z a^{-5} -4 a^{-5} z^{-1} +z^7 a^{-3} +4 z^5 a^{-3} +8 z^3 a^{-3} +11 z a^{-3} +6 a^{-3} z^{-1} -2 z^5 a^{-1} +a z^3-7 z^3 a^{-1} +3 a z-9 z a^{-1} +2 a z^{-1} -5 a^{-1} z^{-1} (db)
Kauffman polynomial -z^{10} a^{-2} -z^{10} a^{-4} -2 z^9 a^{-1} -6 z^9 a^{-3} -4 z^9 a^{-5} -5 z^8 a^{-2} -11 z^8 a^{-4} -8 z^8 a^{-6} -2 z^8-a z^7+z^7 a^{-1} +4 z^7 a^{-3} -7 z^7 a^{-5} -9 z^7 a^{-7} +20 z^6 a^{-2} +28 z^6 a^{-4} +9 z^6 a^{-6} -6 z^6 a^{-8} +7 z^6+5 a z^5+17 z^5 a^{-1} +31 z^5 a^{-3} +37 z^5 a^{-5} +15 z^5 a^{-7} -3 z^5 a^{-9} -13 z^4 a^{-2} -10 z^4 a^{-4} +3 z^4 a^{-6} +6 z^4 a^{-8} -z^4 a^{-10} -7 z^4-9 a z^3-33 z^3 a^{-1} -48 z^3 a^{-3} -41 z^3 a^{-5} -15 z^3 a^{-7} +2 z^3 a^{-9} -z^2 a^{-2} -6 z^2 a^{-4} -9 z^2 a^{-6} -4 z^2 a^{-8} +z^2 a^{-10} +z^2+7 a z+22 z a^{-1} +29 z a^{-3} +20 z a^{-5} +6 z a^{-7} + a^{-2} +3 a^{-4} +3 a^{-6} + a^{-8} +1-2 a z^{-1} -5 a^{-1} z^{-1} -6 a^{-3} z^{-1} -4 a^{-5} z^{-1} - a^{-7} z^{-1} (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
18           1-1
16          2 2
14         51 -4
12        72  5
10       85   -3
8      97    2
6     78     1
4    79      -2
2   59       4
0  15        -4
-2 15         4
-4 1          -1
-61           1
Integral Khovanov Homology

(db, data source)

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