K11a103

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K11a102.gif

K11a102

K11a104.gif

K11a104

Contents

K11a103.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a103 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X14,6,15,5 X12,8,13,7 X18,9,19,10 X2,11,3,12 X6,14,7,13 X22,16,1,15 X20,18,21,17 X8,19,9,20 X16,22,17,21
Gauss code 1, -6, 2, -1, 3, -7, 4, -10, 5, -2, 6, -4, 7, -3, 8, -11, 9, -5, 10, -9, 11, -8
Dowker-Thistlethwaite code 4 10 14 12 18 2 6 22 20 8 16
A Braid Representative
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A Morse Link Presentation K11a103 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:K11a103/ThurstonBennequinNumber
Hyperbolic Volume 13.2218
A-Polynomial See Data:K11a103/A-polynomial

[edit Notes for K11a103's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 0
Rasmussen s-Invariant 0

[edit Notes for K11a103's four dimensional invariants]

Polynomial invariants

Alexander polynomial 4 t^2-20 t+33-20 t^{-1} +4 t^{-2}
Conway polynomial 4 z^4-4 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 81, 0 }
Jones polynomial -q^7+3 q^6-5 q^5+9 q^4-11 q^3+12 q^2-13 q+11-8 q^{-1} +5 q^{-2} -2 q^{-3} + q^{-4}
HOMFLY-PT polynomial (db, data sources) a^4-2 z^2 a^2+z^4-2 z^2-1+2 z^4 a^{-2} +z^2 a^{-2} +z^4 a^{-4} + a^{-4} -z^2 a^{-6}
Kauffman polynomial (db, data sources) z^{10} a^{-2} +z^{10} a^{-4} +3 z^9 a^{-1} +6 z^9 a^{-3} +3 z^9 a^{-5} +5 z^8 a^{-2} +4 z^8 a^{-4} +3 z^8 a^{-6} +4 z^8+4 a z^7-3 z^7 a^{-1} -18 z^7 a^{-3} -10 z^7 a^{-5} +z^7 a^{-7} +3 a^2 z^6-20 z^6 a^{-2} -25 z^6 a^{-4} -13 z^6 a^{-6} -5 z^6+2 a^3 z^5-5 a z^5-2 z^5 a^{-1} +17 z^5 a^{-3} +8 z^5 a^{-5} -4 z^5 a^{-7} +a^4 z^4-2 a^2 z^4+19 z^4 a^{-2} +32 z^4 a^{-4} +17 z^4 a^{-6} +z^4-2 a^3 z^3+5 a z^3+z^3 a^{-1} -13 z^3 a^{-3} -3 z^3 a^{-5} +4 z^3 a^{-7} -2 a^4 z^2-7 z^2 a^{-2} -16 z^2 a^{-4} -7 z^2 a^{-6} +4 z^2-2 a z+2 z a^{-1} +6 z a^{-3} +2 z a^{-5} +a^4+ a^{-4} -1
The A2 invariant Data:K11a103/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a103/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a201,}

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

Vassiliev invariants

V2 and V3: (-4, 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
-16 0 128 \frac{424}{3} \frac{104}{3} 0 -64 -64 0 -\frac{2048}{3} 0 -\frac{6784}{3} -\frac{1664}{3} -\frac{31022}{15} -\frac{1384}{5} -\frac{37448}{45} \frac{686}{9} -\frac{1742}{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=0 is the signature of K11a103. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234567χ
15           1-1
13          2 2
11         31 -2
9        62  4
7       53   -2
5      76    1
3     65     -1
1    57      -2
-1   47       3
-3  14        -3
-5 14         3
-7 1          -1
-91           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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K11a102.gif

K11a102

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K11a104