K11n64

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K11n63.gif

K11n63

K11n65.gif

K11n65

Contents

K11n64.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n64 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X5,15,6,14 X2837 X9,17,10,16 X11,20,12,21 X13,18,14,19 X15,7,16,6 X17,1,18,22 X19,12,20,13 X21,10,22,11
Gauss code 1, -4, 2, -1, -3, 8, 4, -2, -5, 11, -6, 10, -7, 3, -8, 5, -9, 7, -10, 6, -11, 9
Dowker-Thistlethwaite code 4 8 -14 2 -16 -20 -18 -6 -22 -12 -10
A Braid Representative
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A Morse Link Presentation K11n64 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n64/ThurstonBennequinNumber
Hyperbolic Volume 7.23965
A-Polynomial See Data:K11n64/A-polynomial

[edit Notes for K11n64's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -4

[edit Notes for K11n64's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+5 t^2-3 t+1-3 t^{-1} +5 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6-7 z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 21, 4 }
Jones polynomial -q^7+2 q^6-2 q^5+3 q^4-3 q^3+3 q^2-3 q+2- q^{-1} + q^{-2}
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-2} -z^6 a^{-4} -5 z^4 a^{-2} -4 z^4 a^{-4} +z^4 a^{-6} +z^4-7 z^2 a^{-2} -2 z^2 a^{-4} +4 z^2 a^{-6} +4 z^2-4 a^{-2} + a^{-4} +2 a^{-6} - a^{-8} +3
Kauffman polynomial (db, data sources) z^9 a^{-1} +z^9 a^{-3} +3 z^8 a^{-2} +2 z^8 a^{-4} +z^8-6 z^7 a^{-1} -4 z^7 a^{-3} +2 z^7 a^{-5} -18 z^6 a^{-2} -9 z^6 a^{-4} +2 z^6 a^{-6} -7 z^6+11 z^5 a^{-1} +3 z^5 a^{-3} -7 z^5 a^{-5} +z^5 a^{-7} +34 z^4 a^{-2} +11 z^4 a^{-4} -7 z^4 a^{-6} +16 z^4-7 z^3 a^{-1} -z^3 a^{-3} +4 z^3 a^{-5} -2 z^3 a^{-7} -23 z^2 a^{-2} -6 z^2 a^{-4} +6 z^2 a^{-6} +2 z^2 a^{-8} -13 z^2+2 z a^{-1} +2 z a^{-3} +z a^{-7} +z a^{-9} +4 a^{-2} + a^{-4} -2 a^{-6} - a^{-8} +3
The A2 invariant Data:K11n64/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n64/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11n28,}

Vassiliev invariants

V2 and V3: (-1, 1)
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 8 8 \frac{370}{3} \frac{182}{3} -32 \frac{848}{3} \frac{320}{3} 72 -\frac{32}{3} 32 -\frac{1480}{3} -\frac{728}{3} \frac{929}{30} \frac{5542}{15} -\frac{23462}{45} \frac{2911}{18} -\frac{5311}{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=4 is the signature of K11n64. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-1012345χ
15         1-1
13        1 1
11       11 0
9      21  1
7     11   0
5    22    0
3   22     0
1   1      -1
-1 12       1
-3          0
-51         1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=3 i=5
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{2}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=5 {\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 Hoste-Thistlethwaite Knot Page master template (intermediate).

See/edit the Hoste-Thistlethwaite_Splice_Base (expert).

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K11n63.gif

K11n63

K11n65.gif

K11n65