K11a246

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K11a245.gif

K11a245

K11a247.gif

K11a247

Contents

K11a246.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a246 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X14,4,15,3 X20,6,21,5 X22,8,1,7 X18,10,19,9 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, -5, 10, -3, 11, -4
Dowker-Thistlethwaite code 4 14 20 22 18 16 2 12 10 8 6
A Braid Representative
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A Morse Link Presentation K11a246 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:K11a246/ThurstonBennequinNumber
Hyperbolic Volume 8.50058
A-Polynomial See Data:K11a246/A-polynomial

[edit Notes for K11a246'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 K11a246's four dimensional invariants]

Polynomial invariants

Alexander polynomial 4 t^2-10 t+13-10 t^{-1} +4 t^{-2}
Conway polynomial 4 z^4+6 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 41, 4 }
Jones polynomial -q^{13}+2 q^{12}-3 q^{11}+4 q^{10}-5 q^9+6 q^8-6 q^7+5 q^6-4 q^5+3 q^4-q^3+q^2
HOMFLY-PT polynomial (db, data sources) z^4 a^{-4} +z^4 a^{-6} +z^4 a^{-8} +z^4 a^{-10} +3 z^2 a^{-4} +z^2 a^{-6} +z^2 a^{-8} +2 z^2 a^{-10} -z^2 a^{-12} +2 a^{-4} - a^{-6} + 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} +z^7 a^{-7} -3 z^7 a^{-9} -15 z^7 a^{-11} -10 z^7 a^{-13} +z^7 a^{-15} +z^6 a^{-6} -2 z^6 a^{-8} +12 z^6 a^{-10} +5 z^6 a^{-12} -10 z^6 a^{-14} +z^5 a^{-5} -z^5 a^{-7} +4 z^5 a^{-9} +26 z^5 a^{-11} +15 z^5 a^{-13} -5 z^5 a^{-15} +z^4 a^{-4} +3 z^4 a^{-8} -15 z^4 a^{-10} -6 z^4 a^{-12} +13 z^4 a^{-14} -z^3 a^{-5} +2 z^3 a^{-7} -z^3 a^{-9} -20 z^3 a^{-11} -10 z^3 a^{-13} +6 z^3 a^{-15} -3 z^2 a^{-4} -2 z^2 a^{-6} +7 z^2 a^{-10} +4 z^2 a^{-12} -4 z^2 a^{-14} -z a^{-5} -z a^{-7} +4 z a^{-11} +3 z a^{-13} -z a^{-15} +2 a^{-4} + a^{-6} - a^{-10} - a^{-12}
The A2 invariant Data:K11a246/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a246/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_18,}

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

Vassiliev invariants

V2 and V3: (6, 17)
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
24 136 288 876 108 3264 \frac{19024}{3} \frac{2944}{3} 808 2304 9248 21024 2592 \frac{235991}{5} \frac{4556}{5} \frac{261284}{15} \frac{2041}{3} \frac{10711}{5}

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 K11a246. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567891011χ
27           1-1
25          1 1
23         21 -1
21        21  1
19       32   -1
17      32    1
15     33     0
13    23      -1
11   23       1
9  12        -1
7  2         2
511          0
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}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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K11a245.gif

K11a245

K11a247.gif

K11a247