K11a234

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K11a233.gif

K11a233

K11a235.gif

K11a235

Contents

K11a234.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a234 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X14,4,15,3 X16,6,17,5 X18,8,19,7 X20,10,21,9 X22,12,1,11 X2,14,3,13 X12,16,13,15 X6,18,7,17 X8,20,9,19 X10,22,11,21
Gauss code 1, -7, 2, -1, 3, -9, 4, -10, 5, -11, 6, -8, 7, -2, 8, -3, 9, -4, 10, -5, 11, -6
Dowker-Thistlethwaite code 4 14 16 18 20 22 2 12 6 8 10
A Braid Representative
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A Morse Link Presentation K11a234 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 4
3-genus 4
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a234/ThurstonBennequinNumber
Hyperbolic Volume 7.44001
A-Polynomial See Data:K11a234/A-polynomial

[edit Notes for K11a234's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 4
Rasmussen s-Invariant -8

[edit Notes for K11a234's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^4-4 t^3+5 t^2-5 t+5-5 t^{-1} +5 t^{-2} -4 t^{-3} +2 t^{-4}
Conway polynomial 2 z^8+12 z^6+21 z^4+11 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 37, 8 }
Jones polynomial -q^{15}+2 q^{14}-3 q^{13}+4 q^{12}-5 q^{11}+5 q^{10}-5 q^9+4 q^8-3 q^7+3 q^6-q^5+q^4
HOMFLY-PT polynomial (db, data sources) z^8 a^{-8} +z^8 a^{-10} +7 z^6 a^{-8} +6 z^6 a^{-10} -z^6 a^{-12} +16 z^4 a^{-8} +10 z^4 a^{-10} -5 z^4 a^{-12} +14 z^2 a^{-8} +3 z^2 a^{-10} -6 z^2 a^{-12} +4 a^{-8} -2 a^{-10} - a^{-12}
Kauffman polynomial (db, data sources) z^{10} a^{-10} +z^{10} a^{-12} +z^9 a^{-9} +3 z^9 a^{-11} +2 z^9 a^{-13} +z^8 a^{-8} -5 z^8 a^{-10} -4 z^8 a^{-12} +2 z^8 a^{-14} -5 z^7 a^{-9} -16 z^7 a^{-11} -9 z^7 a^{-13} +2 z^7 a^{-15} -7 z^6 a^{-8} +6 z^6 a^{-10} +5 z^6 a^{-12} -6 z^6 a^{-14} +2 z^6 a^{-16} +5 z^5 a^{-9} +25 z^5 a^{-11} +14 z^5 a^{-13} -4 z^5 a^{-15} +2 z^5 a^{-17} +16 z^4 a^{-8} -7 z^4 a^{-12} +5 z^4 a^{-14} -2 z^4 a^{-16} +2 z^4 a^{-18} +3 z^3 a^{-9} -13 z^3 a^{-11} -11 z^3 a^{-13} +3 z^3 a^{-15} -z^3 a^{-17} +z^3 a^{-19} -14 z^2 a^{-8} -3 z^2 a^{-10} +6 z^2 a^{-12} -2 z^2 a^{-14} +z^2 a^{-16} -2 z^2 a^{-18} -3 z a^{-9} +z a^{-11} +2 z a^{-13} -z a^{-15} -z a^{-19} +4 a^{-8} +2 a^{-10} - a^{-12}
The A2 invariant  q^{-14} +2 q^{-18} + q^{-20} +2 q^{-22} + q^{-24} -2 q^{-30} - q^{-34} - q^{-44}
The G2 invariant Data:K11a234/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: (11, 35)
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
44 280 968 \frac{6874}{3} \frac{926}{3} 12320 \frac{62704}{3} \frac{10816}{3} 2328 \frac{42592}{3} 39200 \frac{302456}{3} \frac{40744}{3} \frac{5880341}{30} \frac{55866}{5} \frac{2973682}{45} \frac{18859}{18} \frac{240821}{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=8 is the signature of K11a234. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567891011χ
31           1-1
29          1 1
27         21 -1
25        21  1
23       32   -1
21      22    0
19     33     0
17    12      -1
15   23       1
13  11        0
11  2         2
911          0
71           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=7 i=9
r=0 {\mathbb Z} {\mathbb Z}
r=1 {\mathbb Z}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=7 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=8 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=9 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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K11a233.gif

K11a233

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K11a235