K11n70

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K11n69.gif

K11n69

K11n71.gif

K11n71

Contents

K11n70.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n70 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X5,14,6,15 X2837 X9,19,10,18 X11,21,12,20 X13,6,14,7 X15,22,16,1 X17,13,18,12 X19,11,20,10 X21,16,22,17
Gauss code 1, -4, 2, -1, -3, 7, 4, -2, -5, 10, -6, 9, -7, 3, -8, 11, -9, 5, -10, 6, -11, 8
Dowker-Thistlethwaite code 4 8 -14 2 -18 -20 -6 -22 -12 -10 -16
A Braid Representative
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A Morse Link Presentation K11n70 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n70/ThurstonBennequinNumber
Hyperbolic Volume 6.72199
A-Polynomial See Data:K11n70/A-polynomial

[edit Notes for K11n70's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 2

[edit Notes for K11n70's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+2 t^2-2 t+3-2 t^{-1} +2 t^{-2} - t^{-3}
Conway polynomial -z^6-4 z^4-3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 13, 4 }
Jones polynomial q^6-q^5+q^4-2 q^3+2 q^2-2 q+2- q^{-1} + q^{-2}
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-2} -6 z^4 a^{-2} +z^4 a^{-4} +z^4-11 z^2 a^{-2} +4 z^2 a^{-4} +4 z^2-6 a^{-2} +3 a^{-4} +4
Kauffman polynomial (db, data sources) z^9 a^{-1} +z^9 a^{-3} +3 z^8 a^{-2} +2 z^8 a^{-4} +z^8-6 z^7 a^{-1} -4 z^7 a^{-3} +2 z^7 a^{-5} -19 z^6 a^{-2} -11 z^6 a^{-4} +z^6 a^{-6} -7 z^6+10 z^5 a^{-1} -z^5 a^{-3} -11 z^5 a^{-5} +37 z^4 a^{-2} +16 z^4 a^{-4} -5 z^4 a^{-6} +16 z^4-4 z^3 a^{-1} +11 z^3 a^{-3} +15 z^3 a^{-5} -27 z^2 a^{-2} -9 z^2 a^{-4} +4 z^2 a^{-6} -14 z^2-z a^{-1} -6 z a^{-3} -6 z a^{-5} -z a^{-7} +6 a^{-2} +3 a^{-4} +4
The A2 invariant Data:K11n70/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n70/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {8_1,}

Vassiliev invariants

V2 and V3: (-3, -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
-12 -24 72 114 70 288 528 160 104 -288 288 -1368 -840 -\frac{5471}{10} \frac{11186}{15} -\frac{8114}{5} \frac{597}{2} -\frac{4511}{10}

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 K11n70. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234χ
13        11
11         0
9      22 0
7     1   -1
5    121  0
3   22    0
1   11    0
-1 12      1
-3         0
-51        1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3 i=5
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{2}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z} {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=2 {\mathbb Z} {\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=3 {\mathbb Z}_2 {\mathbb Z}^{2}
r=4 {\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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K11n69.gif

K11n69

K11n71.gif

K11n71