K11a283

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K11a282.gif

K11a282

K11a284.gif

K11a284

Contents

K11a283.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a283 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X10,3,11,4 X16,5,17,6 X14,8,15,7 X18,9,19,10 X2,11,3,12 X20,13,21,14 X22,16,1,15 X4,17,5,18 X12,19,13,20 X8,21,9,22
Gauss code 1, -6, 2, -9, 3, -1, 4, -11, 5, -2, 6, -10, 7, -4, 8, -3, 9, -5, 10, -7, 11, -8
Dowker-Thistlethwaite code 6 10 16 14 18 2 20 22 4 12 8
A Braid Representative
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A Morse Link Presentation K11a283 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a283/ThurstonBennequinNumber
Hyperbolic Volume 17.6295
A-Polynomial See Data:K11a283/A-polynomial

[edit Notes for K11a283's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a283's four dimensional invariants]

Polynomial invariants

Alexander polynomial 3 t^3-15 t^2+34 t-43+34 t^{-1} -15 t^{-2} +3 t^{-3}
Conway polynomial 3 z^6+3 z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 147, -2 }
Jones polynomial -q^2+4 q-9+16 q^{-1} -20 q^{-2} +24 q^{-3} -24 q^{-4} +20 q^{-5} -15 q^{-6} +9 q^{-7} -4 q^{-8} + q^{-9}
HOMFLY-PT polynomial (db, data sources) z^2 a^8+a^8-3 z^4 a^6-6 z^2 a^6-3 a^6+2 z^6 a^4+6 z^4 a^4+7 z^2 a^4+2 a^4+z^6 a^2+z^4 a^2+a^2-z^4-z^2
Kauffman polynomial (db, data sources) z^6 a^{10}-2 z^4 a^{10}+z^2 a^{10}+4 z^7 a^9-9 z^5 a^9+5 z^3 a^9-z a^9+7 z^8 a^8-15 z^6 a^8+8 z^4 a^8-3 z^2 a^8+a^8+7 z^9 a^7-11 z^7 a^7+3 z^3 a^7-z a^7+3 z^{10} a^6+9 z^8 a^6-36 z^6 a^6+37 z^4 a^6-17 z^2 a^6+3 a^6+16 z^9 a^5-36 z^7 a^5+29 z^5 a^5-9 z^3 a^5+2 z a^5+3 z^{10} a^4+13 z^8 a^4-44 z^6 a^4+48 z^4 a^4-20 z^2 a^4+2 a^4+9 z^9 a^3-13 z^7 a^3+8 z^5 a^3-3 z^3 a^3+2 z a^3+11 z^8 a^2-20 z^6 a^2+16 z^4 a^2-6 z^2 a^2-a^2+8 z^7 a-11 z^5 a+3 z^3 a+4 z^6-5 z^4+z^2+z^5 a^{-1} -z^3 a^{-1}
The A2 invariant Data:K11a283/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a283/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: (1, 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
4 0 8 -\frac{130}{3} -\frac{62}{3} 0 224 0 128 \frac{32}{3} 0 -\frac{520}{3} -\frac{248}{3} -\frac{23249}{30} \frac{7298}{15} -\frac{36898}{45} -\frac{2863}{18} -\frac{4049}{30}

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 K11a283. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-8-7-6-5-4-3-2-10123χ
5           1-1
3          3 3
1         61 -5
-1        103  7
-3       117   -4
-5      139    4
-7     1111     0
-9    913      -4
-11   611       5
-13  39        -6
-15 16         5
-17 3          -3
-191           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-8 {\mathbb Z}
r=-7 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-5 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-4 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-3 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=-2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\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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K11a282.gif

K11a282

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K11a284