K11a243

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K11a242.gif

K11a242

K11a244.gif

K11a244

Contents

K11a243.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 X18,6,19,5 X20,8,21,7 X22,10,1,9 X16,12,17,11 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, -6, 9, -3, 10, -4, 11, -5
Dowker-Thistlethwaite code 4 14 18 20 22 16 2 12 10 8 6
A Braid Representative
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A Morse Link Presentation K11a243 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:K11a243/ThurstonBennequinNumber
Hyperbolic Volume 11.4767
A-Polynomial See Data:K11a243/A-polynomial

[edit Notes for K11a243'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 K11a243's four dimensional invariants]

Polynomial invariants

Alexander polynomial 6 t^2-17 t+23-17 t^{-1} +6 t^{-2}
Conway polynomial 6 z^4+7 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 69, 4 }
Jones polynomial -q^{13}+2 q^{12}-4 q^{11}+7 q^{10}-9 q^9+10 q^8-11 q^7+10 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} +2 z^2 a^{-8} +z^2 a^{-10} -z^2 a^{-12} + a^{-4} + a^{-6} - a^{-8} + a^{-10} - 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} -z^7 a^{-9} -12 z^7 a^{-11} -7 z^7 a^{-13} +z^7 a^{-15} +3 z^6 a^{-6} -3 z^6 a^{-8} -3 z^6 a^{-12} -9 z^6 a^{-14} +2 z^5 a^{-5} -z^5 a^{-7} -2 z^5 a^{-9} +12 z^5 a^{-11} +6 z^5 a^{-13} -5 z^5 a^{-15} +z^4 a^{-4} -3 z^4 a^{-6} +z^4 a^{-8} -7 z^4 a^{-10} +12 z^4 a^{-14} -2 z^3 a^{-5} -2 z^3 a^{-7} -9 z^3 a^{-11} -2 z^3 a^{-13} +7 z^3 a^{-15} -2 z^2 a^{-4} +2 z^2 a^{-6} +2 z^2 a^{-8} +5 z^2 a^{-10} +2 z^2 a^{-12} -5 z^2 a^{-14} +2 z a^{-7} +2 z a^{-9} +3 z a^{-11} +z a^{-13} -2 z a^{-15} + a^{-4} - a^{-6} - a^{-8} - a^{-10} - a^{-12}
The A2 invariant Data:K11a243/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a243/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: (7, 20)
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
28 160 392 \frac{3266}{3} \frac{430}{3} 4480 \frac{24928}{3} \frac{4096}{3} 1024 \frac{10976}{3} 12800 \frac{91448}{3} \frac{12040}{3} \frac{1950937}{30} \frac{32366}{15} \frac{1058594}{45} \frac{12359}{18} \frac{87097}{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=4 is the signature of K11a243. 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        41  3
19       53   -2
17      54    1
15     65     -1
13    45      -1
11   36       3
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}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=6 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=7 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=8 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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K11a242.gif

K11a242

K11a244.gif

K11a244