K11n142

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K11n141.gif

K11n141

K11n143.gif

K11n143

Contents

K11n142.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n142 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X12,3,13,4 X5,16,6,17 X7,21,8,20 X9,14,10,15 X11,19,12,18 X2,13,3,14 X15,8,16,9 X17,22,18,1 X19,11,20,10 X21,7,22,6
Gauss code 1, -7, 2, -1, -3, 11, -4, 8, -5, 10, -6, -2, 7, 5, -8, 3, -9, 6, -10, 4, -11, 9
Dowker-Thistlethwaite code 4 12 -16 -20 -14 -18 2 -8 -22 -10 -6
A Braid Representative
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A Morse Link Presentation K11n142 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n142/ThurstonBennequinNumber
Hyperbolic Volume 11.1808
A-Polynomial See Data:K11n142/A-polynomial

[edit Notes for K11n142's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 2
Rasmussen s-Invariant 0

[edit Notes for K11n142's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^2-8 t+15-8 t^{-1} + t^{-2}
Conway polynomial z^4-4 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 33, 0 }
Jones polynomial 2 q^2-3 q+5-6 q^{-1} +5 q^{-2} -5 q^{-3} +4 q^{-4} -2 q^{-5} + q^{-6}
HOMFLY-PT polynomial (db, data sources) a^6-2 z^2 a^4-a^4+z^4 a^2+z^2 a^2+a^2-3 z^2-2+2 a^{-2}
Kauffman polynomial (db, data sources) a^3 z^9+a z^9+2 a^4 z^8+3 a^2 z^8+z^8+2 a^5 z^7-2 a^3 z^7-4 a z^7+a^6 z^6-6 a^4 z^6-11 a^2 z^6-4 z^6-7 a^5 z^5+8 a z^5+z^5 a^{-1} -4 a^6 z^4+2 a^4 z^4+13 a^2 z^4+7 z^4+5 a^5 z^3-3 a^3 z^3-8 a z^3+4 a^6 z^2+a^4 z^2-5 a^2 z^2+2 z^2 a^{-2} -a^5 z+3 a^3 z+5 a z+z a^{-1} -a^6-a^4-a^2-2 a^{-2} -2
The A2 invariant Data:K11n142/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n142/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Vassiliev invariants

V2 and V3: (-4, 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
-16 24 128 \frac{424}{3} \frac{176}{3} -384 -560 -128 -104 -\frac{2048}{3} 288 -\frac{6784}{3} -\frac{2816}{3} -\frac{18782}{15} \frac{1456}{5} -\frac{62288}{45} \frac{2270}{9} -\frac{5102}{15}

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 K11n142. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-6-5-4-3-2-1012χ
5        22
3       1 -1
1      42 2
-1     32  -1
-3    23   -1
-5   33    0
-7  12     -1
-9 13      2
-11 1       -1
-131        1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=2 {\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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K11n141.gif

K11n141

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K11n143