K11a230

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K11a229.gif

K11a229

K11a231.gif

K11a231

Contents

K11a230.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a230 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X12,4,13,3 X20,6,21,5 X18,8,19,7 X16,10,17,9 X2,12,3,11 X22,13,1,14 X10,16,11,15 X8,18,9,17 X6,20,7,19 X14,21,15,22
Gauss code 1, -6, 2, -1, 3, -10, 4, -9, 5, -8, 6, -2, 7, -11, 8, -5, 9, -4, 10, -3, 11, -7
Dowker-Thistlethwaite code 4 12 20 18 16 2 22 10 8 6 14
A Braid Representative
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A Morse Link Presentation K11a230 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 2
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a230/ThurstonBennequinNumber
Hyperbolic Volume 9.27442
A-Polynomial See Data:K11a230/A-polynomial

[edit Notes for K11a230's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a230's four dimensional invariants]

Polynomial invariants

Alexander polynomial -3 t^2+13 t-19+13 t^{-1} -3 t^{-2}
Conway polynomial -3 z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 51, 2 }
Jones polynomial -q^{10}+2 q^9-3 q^8+5 q^7-6 q^6+7 q^5-8 q^4+7 q^3-5 q^2+4 q-2+ q^{-1}
HOMFLY-PT polynomial (db, data sources) -z^4 a^{-2} -z^4 a^{-4} -z^4 a^{-6} -z^2 a^{-2} -z^2 a^{-6} +2 z^2 a^{-8} +z^2- a^{-6} +2 a^{-8} - a^{-10} +1
Kauffman polynomial (db, data sources) z^{10} a^{-6} +z^{10} a^{-8} +2 z^9 a^{-5} +4 z^9 a^{-7} +2 z^9 a^{-9} +2 z^8 a^{-4} -2 z^8 a^{-6} -2 z^8 a^{-8} +2 z^8 a^{-10} +2 z^7 a^{-3} -7 z^7 a^{-5} -19 z^7 a^{-7} -9 z^7 a^{-9} +z^7 a^{-11} +2 z^6 a^{-2} -4 z^6 a^{-4} -2 z^6 a^{-6} -6 z^6 a^{-8} -10 z^6 a^{-10} +2 z^5 a^{-1} -2 z^5 a^{-3} +11 z^5 a^{-5} +31 z^5 a^{-7} +11 z^5 a^{-9} -5 z^5 a^{-11} -z^4 a^{-2} +4 z^4 a^{-4} +7 z^4 a^{-6} +15 z^4 a^{-8} +14 z^4 a^{-10} +z^4-3 z^3 a^{-1} +z^3 a^{-3} -7 z^3 a^{-5} -23 z^3 a^{-7} -6 z^3 a^{-9} +6 z^3 a^{-11} -2 z^2 a^{-2} -5 z^2 a^{-6} -12 z^2 a^{-8} -7 z^2 a^{-10} -2 z^2+3 z a^{-5} +6 z a^{-7} +2 z a^{-9} -z a^{-11} + a^{-6} +2 a^{-8} + a^{-10} +1
The A2 invariant Data:K11a230/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a230/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_36, K11n29,}

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

Vassiliev invariants

V2 and V3: (1, 4)
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 32 8 \frac{494}{3} \frac{82}{3} 128 \frac{2240}{3} \frac{128}{3} 224 \frac{32}{3} 512 \frac{1976}{3} \frac{328}{3} \frac{102751}{30} -\frac{15422}{15} \frac{108542}{45} \frac{4769}{18} \frac{10591}{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 K11a230. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-10123456789χ
21           1-1
19          1 1
17         21 -1
15        31  2
13       32   -1
11      43    1
9     43     -1
7    34      -1
5   24       2
3  23        -1
1 13         2
-1 1          -1
-31           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-2 {\mathbb Z}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=8 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=9 {\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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K11a229.gif

K11a229

K11a231.gif

K11a231