K11a195

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K11a194.gif

K11a194

K11a196.gif

K11a196

Contents

K11a195.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

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Knot presentations

Planar diagram presentation X4251 X12,3,13,4 X14,6,15,5 X20,7,21,8 X18,9,19,10 X16,11,17,12 X2,13,3,14 X22,16,1,15 X10,17,11,18 X8,19,9,20 X6,21,7,22
Gauss code 1, -7, 2, -1, 3, -11, 4, -10, 5, -9, 6, -2, 7, -3, 8, -6, 9, -5, 10, -4, 11, -8
Dowker-Thistlethwaite code 4 12 14 20 18 16 2 22 10 8 6
A Braid Representative
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A Morse Link Presentation K11a195 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 2
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a195/ThurstonBennequinNumber
Hyperbolic Volume 9.31337
A-Polynomial See Data:K11a195/A-polynomial

[edit Notes for K11a195'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 K11a195's four dimensional invariants]

Polynomial invariants

Alexander polynomial 3 t^2-13 t+21-13 t^{-1} +3 t^{-2}
Conway polynomial 3 z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 53, 0 }
Jones polynomial -q^3+3 q^2-4 q+6-7 q^{-1} +8 q^{-2} -7 q^{-3} +6 q^{-4} -5 q^{-5} +3 q^{-6} -2 q^{-7} + q^{-8}
HOMFLY-PT polynomial (db, data sources) a^8-2 z^2 a^6-2 a^6+z^4 a^4+z^2 a^4+a^4+z^4 a^2+z^4+z^2+1-z^2 a^{-2}
Kauffman polynomial (db, data sources) a^6 z^{10}+a^4 z^{10}+2 a^7 z^9+5 a^5 z^9+3 a^3 z^9+a^8 z^8-2 a^6 z^8+a^4 z^8+4 a^2 z^8-12 a^7 z^7-26 a^5 z^7-10 a^3 z^7+4 a z^7-6 a^8 z^6-10 a^6 z^6-19 a^4 z^6-11 a^2 z^6+4 z^6+23 a^7 z^5+42 a^5 z^5+9 a^3 z^5-6 a z^5+4 z^5 a^{-1} +11 a^8 z^4+26 a^6 z^4+29 a^4 z^4+8 a^2 z^4+3 z^4 a^{-2} -3 z^4-17 a^7 z^3-27 a^5 z^3-6 a^3 z^3-3 z^3 a^{-1} +z^3 a^{-3} -7 a^8 z^2-16 a^6 z^2-12 a^4 z^2-3 a^2 z^2-2 z^2 a^{-2} -2 z^2+4 a^7 z+7 a^5 z+3 a^3 z+a^8+2 a^6+a^4+1
The A2 invariant Data:K11a195/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a195/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n114,}

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} -96 112 -64 120 -\frac{32}{3} 288 \frac{1016}{3} \frac{232}{3} \frac{16049}{30} \frac{7222}{15} -\frac{5702}{45} -\frac{2513}{18} -\frac{1231}{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 K11a195. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-8-7-6-5-4-3-2-10123χ
7           1-1
5          2 2
3         21 -1
1        42  2
-1       43   -1
-3      43    1
-5     34     1
-7    34      -1
-9   23       1
-11  13        -2
-13 12         1
-15 1          -1
-171           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-8 {\mathbb Z}
r=-7 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=3 {\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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K11a194

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K11a196