K11a238

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K11a237.gif

K11a237

K11a239.gif

K11a239

Contents

K11a238.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

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Knot presentations

Planar diagram presentation X4251 X14,4,15,3 X16,6,17,5 X20,8,21,7 X18,10,19,9 X22,12,1,11 X2,14,3,13 X12,16,13,15 X10,18,11,17 X8,20,9,19 X6,22,7,21
Gauss code 1, -7, 2, -1, 3, -11, 4, -10, 5, -9, 6, -8, 7, -2, 8, -3, 9, -5, 10, -4, 11, -6
Dowker-Thistlethwaite code 4 14 16 20 18 22 2 12 10 8 6
A Braid Representative
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A Morse Link Presentation K11a238 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 2
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a238/ThurstonBennequinNumber
Hyperbolic Volume 10.8467
A-Polynomial See Data:K11a238/A-polynomial

[edit Notes for K11a238's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a238's four dimensional invariants]

Polynomial invariants

Alexander polynomial 6 t^2-16 t+21-16 t^{-1} +6 t^{-2}
Conway polynomial 6 z^4+8 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 65, 4 }
Jones polynomial -q^{13}+2 q^{12}-4 q^{11}+6 q^{10}-8 q^9+10 q^8-10 q^7+9 q^6-7 q^5+5 q^4-2 q^3+q^2
HOMFLY-PT polynomial (db, data sources) z^4 a^{-4} +2 z^4 a^{-6} +2 z^4 a^{-8} +z^4 a^{-10} +2 z^2 a^{-4} +3 z^2 a^{-6} +3 z^2 a^{-8} +z^2 a^{-10} -z^2 a^{-12} + a^{-4} + a^{-8} - a^{-12}
Kauffman polynomial (db, data sources) z^{10} a^{-10} +z^{10} a^{-12} +2 z^9 a^{-9} +4 z^9 a^{-11} +2 z^9 a^{-13} +3 z^8 a^{-8} -z^8 a^{-12} +2 z^8 a^{-14} +3 z^7 a^{-7} -2 z^7 a^{-9} -13 z^7 a^{-11} -7 z^7 a^{-13} +z^7 a^{-15} +3 z^6 a^{-6} -5 z^6 a^{-8} -4 z^6 a^{-10} -5 z^6 a^{-12} -9 z^6 a^{-14} +2 z^5 a^{-5} -2 z^5 a^{-7} -2 z^5 a^{-9} +11 z^5 a^{-11} +4 z^5 a^{-13} -5 z^5 a^{-15} +z^4 a^{-4} -3 z^4 a^{-6} +7 z^4 a^{-8} +3 z^4 a^{-10} +3 z^4 a^{-12} +11 z^4 a^{-14} -2 z^3 a^{-5} +5 z^3 a^{-9} -2 z^3 a^{-11} +2 z^3 a^{-13} +7 z^3 a^{-15} -2 z^2 a^{-4} +z^2 a^{-6} -5 z^2 a^{-8} -z^2 a^{-10} +3 z^2 a^{-12} -4 z^2 a^{-14} -3 z a^{-9} -z a^{-11} -z a^{-13} -3 z a^{-15} + a^{-4} + a^{-8} - a^{-12}
The A2 invariant Data:K11a238/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a238/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n171,}

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

Vassiliev invariants

V2 and V3: (8, 24)
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
32 192 512 \frac{4192}{3} \frac{608}{3} 6144 11392 1952 1504 \frac{16384}{3} 18432 \frac{134144}{3} \frac{19456}{3} \frac{1418644}{15} \frac{35344}{15} \frac{1636816}{45} \frac{7820}{9} \frac{69844}{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=4 is the signature of K11a238. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567891011χ
27           1-1
25          1 1
23         31 -2
21        31  2
19       53   -2
17      53    2
15     55     0
13    45      -1
11   35       2
9  24        -2
7  3         3
512          -1
31           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=3 i=5
r=0 {\mathbb Z} {\mathbb Z}
r=1 {\mathbb Z}^{2}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=6 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=7 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=8 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=9 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=10 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=11 {\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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K11a237.gif

K11a237

K11a239.gif

K11a239