K11a241

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K11a240.gif

K11a240

K11a242.gif

K11a242

Contents

K11a241.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a241 at Knotilus!



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 X10,16,11,15 X12,18,13,17 X8,20,9,19 X6,22,7,21
Gauss code 1, -7, 2, -1, 3, -11, 4, -10, 5, -8, 6, -9, 7, -2, 8, -6, 9, -3, 10, -4, 11, -5
Dowker-Thistlethwaite code 4 14 18 20 22 16 2 10 12 8 6
A Braid Representative
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A Morse Link Presentation K11a241 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 3
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a241/ThurstonBennequinNumber
Hyperbolic Volume 13.7014
A-Polynomial See Data:K11a241/A-polynomial

[edit Notes for K11a241's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -6

[edit Notes for K11a241's four dimensional invariants]

Polynomial invariants

Alexander polynomial 4 t^3-12 t^2+20 t-23+20 t^{-1} -12 t^{-2} +4 t^{-3}
Conway polynomial 4 z^6+12 z^4+8 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 95, 6 }
Jones polynomial -q^{14}+3 q^{13}-6 q^{12}+10 q^{11}-14 q^{10}+15 q^9-15 q^8+13 q^7-9 q^6+6 q^5-2 q^4+q^3
HOMFLY-PT polynomial (db, data sources) z^6 a^{-6} +2 z^6 a^{-8} +z^6 a^{-10} +4 z^4 a^{-6} +7 z^4 a^{-8} +2 z^4 a^{-10} -z^4 a^{-12} +5 z^2 a^{-6} +6 z^2 a^{-8} -z^2 a^{-10} -2 z^2 a^{-12} +2 a^{-6} + a^{-8} -2 a^{-10}
Kauffman polynomial (db, data sources) z^{10} a^{-10} +z^{10} a^{-12} +2 z^9 a^{-9} +5 z^9 a^{-11} +3 z^9 a^{-13} +3 z^8 a^{-8} +3 z^8 a^{-10} +5 z^8 a^{-12} +5 z^8 a^{-14} +2 z^7 a^{-7} +z^7 a^{-9} -6 z^7 a^{-11} +5 z^7 a^{-15} +z^6 a^{-6} -7 z^6 a^{-8} -6 z^6 a^{-10} -10 z^6 a^{-12} -9 z^6 a^{-14} +3 z^6 a^{-16} -5 z^5 a^{-7} -11 z^5 a^{-9} -2 z^5 a^{-11} -8 z^5 a^{-13} -11 z^5 a^{-15} +z^5 a^{-17} -4 z^4 a^{-6} +5 z^4 a^{-8} +2 z^4 a^{-10} +6 z^4 a^{-12} +7 z^4 a^{-14} -6 z^4 a^{-16} +2 z^3 a^{-7} +10 z^3 a^{-9} +9 z^3 a^{-11} +12 z^3 a^{-13} +9 z^3 a^{-15} -2 z^3 a^{-17} +5 z^2 a^{-6} -3 z^2 a^{-8} -4 z^2 a^{-10} +z^2 a^{-12} -z^2 a^{-14} +2 z^2 a^{-16} +z a^{-7} -4 z a^{-9} -6 z a^{-11} -4 z a^{-13} -3 z a^{-15} -2 a^{-6} + a^{-8} +2 a^{-10}
The A2 invariant Data:K11a241/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a241/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a186,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a186,}

Vassiliev invariants

V2 and V3: (8, 22)
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
32 176 512 \frac{3664}{3} \frac{464}{3} 5632 \frac{28544}{3} \frac{4640}{3} 1072 \frac{16384}{3} 15488 \frac{117248}{3} \frac{14848}{3} \frac{1146124}{15} \frac{18528}{5} \frac{1151536}{45} \frac{6020}{9} \frac{46204}{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=6 is the signature of K11a241. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567891011χ
29           1-1
27          2 2
25         41 -3
23        62  4
21       84   -4
19      76    1
17     88     0
15    57      -2
13   48       4
11  25        -3
9  4         4
712          -1
51           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=5 i=7
r=0 {\mathbb Z} {\mathbb Z}
r=1 {\mathbb Z}^{2}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=6 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=7 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=8 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=9 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=10 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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K11a240.gif

K11a240

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K11a242