K11n68

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K11n67.gif

K11n67

K11n69.gif

K11n69

Contents

K11n68.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n68 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X5,15,6,14 X2837 X18,9,19,10 X16,11,17,12 X20,14,21,13 X15,7,16,6 X10,17,11,18 X22,19,1,20 X12,22,13,21
Gauss code 1, -4, 2, -1, -3, 8, 4, -2, 5, -9, 6, -11, 7, 3, -8, -6, 9, -5, 10, -7, 11, -10
Dowker-Thistlethwaite code 4 8 -14 2 18 16 20 -6 10 22 12
A Braid Representative
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A Morse Link Presentation K11n68 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n68/ThurstonBennequinNumber
Hyperbolic Volume 13.3364
A-Polynomial See Data:K11n68/A-polynomial

[edit Notes for K11n68's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus [1,2]
Rasmussen s-Invariant -2

[edit Notes for K11n68's four dimensional invariants]

Polynomial invariants

Alexander polynomial -4 t^2+16 t-23+16 t^{-1} -4 t^{-2}
Conway polynomial 1-4 z^4
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 63, 2 }
Jones polynomial -q^8+3 q^7-6 q^6+9 q^5-10 q^4+11 q^3-10 q^2+7 q-4+2 q^{-1}
HOMFLY-PT polynomial (db, data sources) -2 z^4 a^{-2} -2 z^4 a^{-4} -3 z^2 a^{-2} -2 z^2 a^{-4} +3 z^2 a^{-6} +2 z^2-2 a^{-2} +2 a^{-6} - a^{-8} +2
Kauffman polynomial (db, data sources) z^9 a^{-3} +z^9 a^{-5} +2 z^8 a^{-2} +5 z^8 a^{-4} +3 z^8 a^{-6} +z^7 a^{-1} +2 z^7 a^{-3} +5 z^7 a^{-5} +4 z^7 a^{-7} -4 z^6 a^{-2} -8 z^6 a^{-4} -z^6 a^{-6} +3 z^6 a^{-8} +z^5 a^{-1} -4 z^5 a^{-3} -12 z^5 a^{-5} -6 z^5 a^{-7} +z^5 a^{-9} +10 z^4 a^{-2} +6 z^4 a^{-4} -7 z^4 a^{-6} -6 z^4 a^{-8} +3 z^4-3 z^3 a^{-1} +4 z^3 a^{-3} +9 z^3 a^{-5} -2 z^3 a^{-9} -10 z^2 a^{-2} -z^2 a^{-4} +6 z^2 a^{-6} +3 z^2 a^{-8} -6 z^2+z a^{-1} -2 z a^{-5} +z a^{-9} +2 a^{-2} -2 a^{-6} - a^{-8} +2
The A2 invariant Data:K11n68/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n68/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_67, 10_74,}

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

Vassiliev invariants

V2 and V3: (0, 2)
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
0 16 0 96 32 0 \frac{832}{3} \frac{160}{3} 80 0 128 0 0 784 -\frac{208}{3} \frac{1312}{3} \frac{160}{3} 32

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 K11n68. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-101234567χ
17         1-1
15        2 2
13       41 -3
11      52  3
9     54   -1
7    65    1
5   45     1
3  36      -3
1 25       3
-1 2        -2
-32         2
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-2 {\mathbb Z}^{2}
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}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=7 {\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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K11n67.gif

K11n67

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K11n69