K11n51

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K11n50.gif

K11n50

K11n52.gif

K11n52

Contents

K11n51.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n51 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X14,6,15,5 X2837 X9,16,10,17 X11,19,12,18 X6,14,7,13 X15,22,16,1 X17,20,18,21 X19,11,20,10 X21,12,22,13
Gauss code 1, -4, 2, -1, 3, -7, 4, -2, -5, 10, -6, 11, 7, -3, -8, 5, -9, 6, -10, 9, -11, 8
Dowker-Thistlethwaite code 4 8 14 2 -16 -18 6 -22 -20 -10 -12
A Braid Representative
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A Morse Link Presentation K11n51 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:K11n51/ThurstonBennequinNumber
Hyperbolic Volume 9.50422
A-Polynomial See Data:K11n51/A-polynomial

[edit Notes for K11n51's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 0

[edit Notes for K11n51's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+4 t^2-6 t+7-6 t^{-1} +4 t^{-2} - t^{-3}
Conway polynomial -z^6-2 z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 29, 0 }
Jones polynomial -q^7+2 q^6-3 q^5+4 q^4-4 q^3+5 q^2-4 q+3-2 q^{-1} + q^{-2}
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-2} -5 z^4 a^{-2} +2 z^4 a^{-4} +z^4-8 z^2 a^{-2} +7 z^2 a^{-4} -z^2 a^{-6} +3 z^2-4 a^{-2} +5 a^{-4} -2 a^{-6} +2
Kauffman polynomial (db, data sources) z^9 a^{-3} +z^9 a^{-5} +2 z^8 a^{-2} +4 z^8 a^{-4} +2 z^8 a^{-6} +z^7 a^{-1} -2 z^7 a^{-3} -2 z^7 a^{-5} +z^7 a^{-7} -9 z^6 a^{-2} -19 z^6 a^{-4} -10 z^6 a^{-6} -2 z^5 a^{-1} -4 z^5 a^{-3} -7 z^5 a^{-5} -5 z^5 a^{-7} +16 z^4 a^{-2} +27 z^4 a^{-4} +14 z^4 a^{-6} +3 z^4+2 a z^3+2 z^3 a^{-1} +7 z^3 a^{-3} +14 z^3 a^{-5} +7 z^3 a^{-7} +a^2 z^2-15 z^2 a^{-2} -17 z^2 a^{-4} -7 z^2 a^{-6} -4 z^2-a z-2 z a^{-1} -3 z a^{-3} -5 z a^{-5} -3 z a^{-7} +4 a^{-2} +5 a^{-4} +2 a^{-6} +2
The A2 invariant Data:K11n51/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n51/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_127, 10_150,}

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

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{254}{3} \frac{58}{3} 96 272 32 56 \frac{32}{3} 288 \frac{1016}{3} \frac{232}{3} \frac{34831}{30} -\frac{742}{15} \frac{24422}{45} \frac{497}{18} \frac{1711}{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=0 is the signature of K11n51. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-101234567χ
15         1-1
13        1 1
11       21 -1
9      21  1
7     22   0
5    32    1
3   12     1
1  23      -1
-1 12       1
-3 1        -1
-51         1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-2 {\mathbb Z}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
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 Hoste-Thistlethwaite Knot Page master template (intermediate).

See/edit the Hoste-Thistlethwaite_Splice_Base (expert).

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K11n50.gif

K11n50

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K11n52