K11n170

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K11n169.gif

K11n169

K11n171.gif

K11n171

Contents

K11n170.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n170 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X10,3,11,4 X16,5,17,6 X12,8,13,7 X18,9,19,10 X2,11,3,12 X13,22,14,1 X15,20,16,21 X4,17,5,18 X8,19,9,20 X21,14,22,15
Gauss code 1, -6, 2, -9, 3, -1, 4, -10, 5, -2, 6, -4, -7, 11, -8, -3, 9, -5, 10, 8, -11, 7
Dowker-Thistlethwaite code 6 10 16 12 18 2 -22 -20 4 8 -14
A Braid Representative
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A Morse Link Presentation K11n170 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:K11n170/ThurstonBennequinNumber
Hyperbolic Volume 13.6098
A-Polynomial See Data:K11n170/A-polynomial

[edit Notes for K11n170's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n170's four dimensional invariants]

Polynomial invariants

Alexander polynomial -3 t^2+16 t-25+16 t^{-1} -3 t^{-2}
Conway polynomial -3 z^4+4 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 63, -2 }
Jones polynomial q-3+6 q^{-1} -9 q^{-2} +11 q^{-3} -10 q^{-4} +10 q^{-5} -7 q^{-6} +4 q^{-7} -2 q^{-8}
HOMFLY-PT polynomial (db, data sources) -2 a^8+4 z^2 a^6+3 a^6-2 z^4 a^4-2 z^2 a^4-a^4-z^4 a^2+z^2 a^2+a^2+z^2
Kauffman polynomial (db, data sources) 3 z^5 a^9-8 z^3 a^9+5 z a^9+z^8 a^8-4 z^4 a^8+3 z^2 a^8-2 a^8+z^9 a^7+z^5 a^7-9 z^3 a^7+5 z a^7+5 z^8 a^6-9 z^6 a^6+3 z^4 a^6+2 z^2 a^6-3 a^6+z^9 a^5+6 z^7 a^5-15 z^5 a^5+10 z^3 a^5-z a^5+4 z^8 a^4-4 z^6 a^4+3 z^2 a^4-a^4+6 z^7 a^3-10 z^5 a^3+8 z^3 a^3-z a^3+5 z^6 a^2-6 z^4 a^2+3 z^2 a^2-a^2+3 z^5 a-3 z^3 a+z^4-z^2
The A2 invariant Data:K11n170/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n170/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, -9)
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 -72 128 \frac{1208}{3} \frac{256}{3} -1152 -2416 -384 -520 \frac{2048}{3} 2592 \frac{19328}{3} \frac{4096}{3} \frac{216302}{15} -\frac{20608}{15} \frac{361088}{45} \frac{2194}{9} \frac{18782}{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=-2 is the signature of K11n170. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-7-6-5-4-3-2-1012χ
3         11
1        2 -2
-1       41 3
-3      63  -3
-5     53   2
-7    56    1
-9   55     0
-11  25      3
-13 25       -3
-15 2        2
-172         -2
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-7 {\mathbb Z}^{2}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\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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K11n169

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K11n171