K11a239

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K11a238.gif

K11a238

K11a240.gif

K11a240

Contents

K11a239.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a239 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X14,4,15,3 X18,5,19,6 X20,8,21,7 X22,9,1,10 X6,12,7,11 X2,14,3,13 X8,15,9,16 X10,18,11,17 X12,19,13,20 X16,22,17,21
Gauss code 1, -7, 2, -1, 3, -6, 4, -8, 5, -9, 6, -10, 7, -2, 8, -11, 9, -3, 10, -4, 11, -5
Dowker-Thistlethwaite code 4 14 18 20 22 6 2 8 10 12 16
A Braid Representative
BraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation K11a239 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a239/ThurstonBennequinNumber
Hyperbolic Volume 19.5737
A-Polynomial See Data:K11a239/A-polynomial

[edit Notes for K11a239's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a239's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+7 t^3-22 t^2+42 t-51+42 t^{-1} -22 t^{-2} +7 t^{-3} - t^{-4}
Conway polynomial -z^8-z^6+z^2+1
2nd Alexander ideal (db, data sources) \left\{3,t^2+1\right\}
Determinant and Signature { 195, 2 }
Jones polynomial -q^8+5 q^7-12 q^6+20 q^5-28 q^4+32 q^3-31 q^2+28 q-20+12 q^{-1} -5 q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) -z^8 a^{-2} -4 z^6 a^{-2} +2 z^6 a^{-4} +z^6-6 z^4 a^{-2} +5 z^4 a^{-4} -z^4 a^{-6} +2 z^4-2 z^2 a^{-2} +3 z^2 a^{-4} -z^2 a^{-6} +z^2+2 a^{-2} - a^{-4}
Kauffman polynomial (db, data sources) 4 z^{10} a^{-2} +4 z^{10} a^{-4} +11 z^9 a^{-1} +24 z^9 a^{-3} +13 z^9 a^{-5} +23 z^8 a^{-2} +29 z^8 a^{-4} +17 z^8 a^{-6} +11 z^8+5 a z^7-12 z^7 a^{-1} -33 z^7 a^{-3} -4 z^7 a^{-5} +12 z^7 a^{-7} +a^2 z^6-66 z^6 a^{-2} -72 z^6 a^{-4} -24 z^6 a^{-6} +5 z^6 a^{-8} -22 z^6-8 a z^5-8 z^5 a^{-1} -6 z^5 a^{-3} -21 z^5 a^{-5} -14 z^5 a^{-7} +z^5 a^{-9} -a^2 z^4+45 z^4 a^{-2} +45 z^4 a^{-4} +13 z^4 a^{-6} -3 z^4 a^{-8} +15 z^4+4 a z^3+8 z^3 a^{-1} +10 z^3 a^{-3} +12 z^3 a^{-5} +6 z^3 a^{-7} -8 z^2 a^{-2} -8 z^2 a^{-4} -3 z^2 a^{-6} -3 z^2+z a^{-3} +z a^{-5} -2 a^{-2} - a^{-4}
The A2 invariant q^8-3 q^6+4 q^4-3 q^2-1+6 q^{-2} -5 q^{-4} +8 q^{-6} -3 q^{-8} + q^{-10} + q^{-12} -6 q^{-14} +5 q^{-16} -3 q^{-18} +2 q^{-22} - q^{-24}
The G2 invariant Data:K11a239/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, 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
4 8 8 \frac{62}{3} \frac{10}{3} 32 \frac{176}{3} -\frac{64}{3} 40 \frac{32}{3} 32 \frac{248}{3} \frac{40}{3} \frac{5071}{30} -\frac{1382}{15} \frac{5942}{45} \frac{593}{18} \frac{271}{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 K11a239. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234567χ
17           1-1
15          4 4
13         81 -7
11        124  8
9       168   -8
7      1612    4
5     1516     1
3    1316      -3
1   816       8
-1  412        -8
-3 18         7
-5 4          -4
-71           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{16}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{13}
r=1 {\mathbb Z}^{16}\oplus{\mathbb Z}_2^{15} {\mathbb Z}^{15}
r=2 {\mathbb Z}^{16}\oplus{\mathbb Z}_2^{16} {\mathbb Z}^{16}
r=3 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{16} {\mathbb Z}^{16}
r=4 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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K11a238.gif

K11a238

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K11a240