K11a278

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K11a277.gif

K11a277

K11a279.gif

K11a279

Contents

K11a278.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a278 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X10,4,11,3 X16,5,17,6 X12,8,13,7 X4,10,5,9 X18,11,19,12 X22,14,1,13 X20,16,21,15 X2,17,3,18 X8,19,9,20 X14,22,15,21
Gauss code 1, -9, 2, -5, 3, -1, 4, -10, 5, -2, 6, -4, 7, -11, 8, -3, 9, -6, 10, -8, 11, -7
Dowker-Thistlethwaite code 6 10 16 12 4 18 22 20 2 8 14
A Braid Representative
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A Morse Link Presentation K11a278 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a278/ThurstonBennequinNumber
Hyperbolic Volume 17.2112
A-Polynomial See Data:K11a278/A-polynomial

[edit Notes for K11a278's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 0

[edit Notes for K11a278's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+12 t^2-33 t+47-33 t^{-1} +12 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6-3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 141, 0 }
Jones polynomial -q^7+4 q^6-8 q^5+14 q^4-19 q^3+22 q^2-23 q+20-15 q^{-1} +10 q^{-2} -4 q^{-3} + q^{-4}
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-2} -z^6+a^2 z^4-z^4 a^{-2} +2 z^4 a^{-4} -2 z^4+a^2 z^2-z^2 a^{-2} +2 z^2 a^{-4} -z^2 a^{-6} -4 z^2+2 a^2+ a^{-4} -2
Kauffman polynomial (db, data sources) 3 z^{10} a^{-2} +3 z^{10} a^{-4} +10 z^9 a^{-1} +16 z^9 a^{-3} +6 z^9 a^{-5} +15 z^8 a^{-2} +4 z^8 a^{-4} +4 z^8 a^{-6} +15 z^8+15 a z^7-9 z^7 a^{-1} -44 z^7 a^{-3} -19 z^7 a^{-5} +z^7 a^{-7} +10 a^2 z^6-54 z^6 a^{-2} -37 z^6 a^{-4} -14 z^6 a^{-6} -21 z^6+4 a^3 z^5-20 a z^5-9 z^5 a^{-1} +35 z^5 a^{-3} +17 z^5 a^{-5} -3 z^5 a^{-7} +a^4 z^4-9 a^2 z^4+44 z^4 a^{-2} +46 z^4 a^{-4} +15 z^4 a^{-6} +3 z^4+8 a z^3-14 z^3 a^{-3} -4 z^3 a^{-5} +2 z^3 a^{-7} +5 a^2 z^2-13 z^2 a^{-2} -17 z^2 a^{-4} -5 z^2 a^{-6} +4 z^2+4 z a^{-1} +5 z a^{-3} +z a^{-5} -2 a^2+ a^{-4} -2
The A2 invariant Data:K11a278/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a278/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: (-3, -1)
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 -8 72 98 38 96 \frac{400}{3} -\frac{32}{3} 56 -288 32 -1176 -456 -\frac{10351}{10} -\frac{94}{15} -\frac{12422}{15} \frac{1231}{6} -\frac{1711}{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=0 is the signature of K11a278. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234567χ
15           1-1
13          3 3
11         51 -4
9        93  6
7       105   -5
5      129    3
3     1110     -1
1    912      -3
-1   712       5
-3  38        -5
-5 17         6
-7 3          -3
-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}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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K11a277.gif

K11a277

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K11a279