K11n158

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K11n157.gif

K11n157

K11n159.gif

K11n159

Contents

K11n158.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n158 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X10,3,11,4 X12,6,13,5 X18,8,19,7 X20,10,21,9 X16,11,17,12 X13,1,14,22 X4,16,5,15 X2,17,3,18 X8,20,9,19 X21,15,22,14
Gauss code 1, -9, 2, -8, 3, -1, 4, -10, 5, -2, 6, -3, -7, 11, 8, -6, 9, -4, 10, -5, -11, 7
Dowker-Thistlethwaite code 6 10 12 18 20 16 -22 4 2 8 -14
A Braid Representative
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A Morse Link Presentation K11n158 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n158/ThurstonBennequinNumber
Hyperbolic Volume 12.7254
A-Polynomial See Data:K11n158/A-polynomial

[edit Notes for K11n158's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n158's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-4 t^3+7 t^2-7 t+7-7 t^{-1} +7 t^{-2} -4 t^{-3} + t^{-4}
Conway polynomial z^8+4 z^6+3 z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 45, 4 }
Jones polynomial q^8-3 q^7+5 q^6-7 q^5+7 q^4-7 q^3+7 q^2-4 q+3- q^{-1}
HOMFLY-PT polynomial (db, data sources) z^8 a^{-4} -z^6 a^{-2} +6 z^6 a^{-4} -z^6 a^{-6} -4 z^4 a^{-2} +12 z^4 a^{-4} -5 z^4 a^{-6} -3 z^2 a^{-2} +10 z^2 a^{-4} -7 z^2 a^{-6} +z^2 a^{-8} + a^{-2} +2 a^{-4} -3 a^{-6} + a^{-8}
Kauffman polynomial (db, data sources) 2 z^9 a^{-3} +2 z^9 a^{-5} +3 z^8 a^{-2} +7 z^8 a^{-4} +4 z^8 a^{-6} +z^7 a^{-1} -6 z^7 a^{-3} -4 z^7 a^{-5} +3 z^7 a^{-7} -14 z^6 a^{-2} -29 z^6 a^{-4} -14 z^6 a^{-6} +z^6 a^{-8} -4 z^5 a^{-1} +z^5 a^{-3} -2 z^5 a^{-5} -7 z^5 a^{-7} +19 z^4 a^{-2} +35 z^4 a^{-4} +17 z^4 a^{-6} +z^4 a^{-8} +4 z^3 a^{-1} +2 z^3 a^{-3} +5 z^3 a^{-7} +3 z^3 a^{-9} -7 z^2 a^{-2} -17 z^2 a^{-4} -12 z^2 a^{-6} -z^2 a^{-8} +z^2 a^{-10} +2 z a^{-3} +2 z a^{-5} -z a^{-7} -z a^{-9} - a^{-2} +2 a^{-4} +3 a^{-6} + a^{-8}
The A2 invariant -q^2+1+2 q^{-4} +2 q^{-6} + q^{-8} +2 q^{-10} -3 q^{-12} + q^{-14} -2 q^{-16} - q^{-22} + q^{-24} - q^{-26} + q^{-28}
The G2 invariant Data:K11n158/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: (1, 0)
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 0 8 -\frac{130}{3} -\frac{62}{3} 0 -224 -32 -96 \frac{32}{3} 0 -\frac{520}{3} -\frac{248}{3} -\frac{23249}{30} \frac{2978}{15} -\frac{25378}{45} -\frac{2287}{18} -\frac{2129}{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 K11n158. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-3-2-10123456χ
17         11
15        2 -2
13       31 2
11      42  -2
9     33   0
7    44    0
5   33     0
3  25      3
1 12       -1
-1 2        2
-31         -1
Integral Khovanov Homology

(db, data source)

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

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K11n157.gif

K11n157

K11n159.gif

K11n159