K11n66

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K11n65.gif

K11n65

K11n67.gif

K11n67

Contents

K11n66.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n66 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X14,5,15,6 X2837 X9,19,10,18 X11,17,12,16 X20,14,21,13 X6,15,7,16 X17,11,18,10 X22,19,1,20 X12,22,13,21
Gauss code 1, -4, 2, -1, 3, -8, 4, -2, -5, 9, -6, -11, 7, -3, 8, 6, -9, 5, 10, -7, 11, -10
Dowker-Thistlethwaite code 4 8 14 2 -18 -16 20 6 -10 22 12
A Braid Representative
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A Morse Link Presentation K11n66 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n66/ThurstonBennequinNumber
Hyperbolic Volume 14.4341
A-Polynomial See Data:K11n66/A-polynomial

[edit Notes for K11n66's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus [1,3]
Rasmussen s-Invariant -2

[edit Notes for K11n66's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-7 t^2+18 t-23+18 t^{-1} -7 t^{-2} + t^{-3}
Conway polynomial z^6-z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 75, 2 }
Jones polynomial 2 q^7-5 q^6+8 q^5-12 q^4+13 q^3-12 q^2+11 q-7+4 q^{-1} - q^{-2}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +3 z^4 a^{-2} -3 z^4 a^{-4} -z^4+5 z^2 a^{-2} -7 z^2 a^{-4} +2 z^2 a^{-6} -z^2+4 a^{-2} -5 a^{-4} +2 a^{-6}
Kauffman polynomial (db, data sources) z^9 a^{-3} +z^9 a^{-5} +4 z^8 a^{-2} +6 z^8 a^{-4} +2 z^8 a^{-6} +6 z^7 a^{-1} +10 z^7 a^{-3} +5 z^7 a^{-5} +z^7 a^{-7} +z^6 a^{-2} -3 z^6 a^{-4} +4 z^6+a z^5-10 z^5 a^{-1} -21 z^5 a^{-3} -7 z^5 a^{-5} +3 z^5 a^{-7} -15 z^4 a^{-2} -11 z^4 a^{-4} +3 z^4 a^{-8} -7 z^4-a z^3+3 z^3 a^{-1} +10 z^3 a^{-3} -6 z^3 a^{-7} +12 z^2 a^{-2} +14 z^2 a^{-4} +z^2 a^{-6} -4 z^2 a^{-8} +3 z^2-z a^{-1} +4 z a^{-5} +3 z a^{-7} -4 a^{-2} -5 a^{-4} -2 a^{-6}
The A2 invariant Data:K11n66/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n66/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_40, 10_59,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11n150,}

Vassiliev invariants

V2 and V3: (-1, -3)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9
-4 -24 8 -\frac{110}{3} \frac{38}{3} 96 144 64 72 -\frac{32}{3} 288 \frac{440}{3} -\frac{152}{3} \frac{29249}{30} -\frac{1018}{15} \frac{26218}{45} \frac{1183}{18} \frac{1409}{30}

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r). The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s-1, where s=2 is the signature of K11n66. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-3-2-10123456χ
15         22
13        3 -3
11       52 3
9      73  -4
7     65   1
5    67    1
3   56     -1
1  37      4
-1 14       -3
-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}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
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}_2^{2} {\mathbb Z}^{2}

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 Hoste-Thistlethwaite Knot Page master template (intermediate).

See/edit the Hoste-Thistlethwaite_Splice_Base (expert).

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K11n65.gif

K11n65

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K11n67