K11a280

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K11a279.gif

K11a279

K11a281.gif

K11a281

Contents

K11a280.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a280 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 X22,14,1,13 X20,16,21,15 X4,17,5,18 X8,19,9,20 X14,22,15,21
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 K11a280 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a280/ThurstonBennequinNumber
Hyperbolic Volume 15.311
A-Polynomial See Data:K11a280/A-polynomial

[edit Notes for K11a280'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 K11a280's four dimensional invariants]

Polynomial invariants

Alexander polynomial 6 t^2-26 t+41-26 t^{-1} +6 t^{-2}
Conway polynomial 6 z^4-2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 105, 0 }
Jones polynomial -q^5+3 q^4-6 q^3+11 q^2-14 q+17-17 q^{-1} +14 q^{-2} -11 q^{-3} +7 q^{-4} -3 q^{-5} + q^{-6}
HOMFLY-PT polynomial (db, data sources) a^6-3 z^2 a^4-a^4+2 z^4 a^2-a^2+3 z^4+3 z^2+2+z^4 a^{-2} -z^2 a^{-2} -z^2 a^{-4}
Kauffman polynomial (db, data sources) 2 a^2 z^{10}+2 z^{10}+4 a^3 z^9+10 a z^9+6 z^9 a^{-1} +4 a^4 z^8+2 a^2 z^8+7 z^8 a^{-2} +5 z^8+3 a^5 z^7-7 a^3 z^7-32 a z^7-17 z^7 a^{-1} +5 z^7 a^{-3} +a^6 z^6-7 a^4 z^6-15 a^2 z^6-20 z^6 a^{-2} +3 z^6 a^{-4} -30 z^6-8 a^5 z^5+2 a^3 z^5+44 a z^5+23 z^5 a^{-1} -10 z^5 a^{-3} +z^5 a^{-5} -3 a^6 z^4-3 a^4 z^4+19 a^2 z^4+25 z^4 a^{-2} -6 z^4 a^{-4} +50 z^4+5 a^5 z^3-5 a^3 z^3-26 a z^3-10 z^3 a^{-1} +4 z^3 a^{-3} -2 z^3 a^{-5} +3 a^6 z^2+4 a^4 z^2-11 a^2 z^2-11 z^2 a^{-2} +z^2 a^{-4} -24 z^2-a^5 z+3 a^3 z+7 a z+3 z a^{-1} -a^6-a^4+a^2+2
The A2 invariant Data:K11a280/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a280/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a16,}

Vassiliev invariants

V2 and V3: (-2, 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
-8 24 32 -\frac{124}{3} -\frac{92}{3} -192 -176 -128 88 -\frac{256}{3} 288 \frac{992}{3} \frac{736}{3} \frac{14369}{15} -\frac{956}{15} \frac{35996}{45} -\frac{1601}{9} \frac{2129}{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 K11a280. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-6-5-4-3-2-1012345χ
11           1-1
9          2 2
7         41 -3
5        72  5
3       74   -3
1      107    3
-1     88     0
-3    69      -3
-5   58       3
-7  26        -4
-9 15         4
-11 2          -2
-131           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=5 {\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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K11a279.gif

K11a279

K11a281.gif

K11a281