K11a273

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K11a272.gif

K11a272

K11a274.gif

K11a274

Contents

K11a273.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a273 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X10,3,11,4 X14,5,15,6 X16,8,17,7 X20,9,21,10 X4,11,5,12 X18,13,19,14 X2,15,3,16 X22,18,1,17 X12,19,13,20 X8,21,9,22
Gauss code 1, -8, 2, -6, 3, -1, 4, -11, 5, -2, 6, -10, 7, -3, 8, -4, 9, -7, 10, -5, 11, -9
Dowker-Thistlethwaite code 6 10 14 16 20 4 18 2 22 12 8
A Braid Representative
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A Morse Link Presentation K11a273 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a273/ThurstonBennequinNumber
Hyperbolic Volume 18.4008
A-Polynomial See Data:K11a273/A-polynomial

[edit Notes for K11a273's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 2

[edit Notes for K11a273's four dimensional invariants]

Polynomial invariants

Alexander polynomial 3 t^3-16 t^2+37 t-47+37 t^{-1} -16 t^{-2} +3 t^{-3}
Conway polynomial 3 z^6+2 z^4+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 159, -2 }
Jones polynomial -q^2+4 q-9+17 q^{-1} -22 q^{-2} +26 q^{-3} -26 q^{-4} +22 q^{-5} -17 q^{-6} +10 q^{-7} -4 q^{-8} + q^{-9}
HOMFLY-PT polynomial (db, data sources) z^2 a^8+a^8-3 z^4 a^6-5 z^2 a^6-2 a^6+2 z^6 a^4+5 z^4 a^4+4 z^2 a^4+z^6 a^2+z^4 a^2+z^2 a^2+2 a^2-z^4-z^2
Kauffman polynomial (db, data sources) z^6 a^{10}-2 z^4 a^{10}+z^2 a^{10}+4 z^7 a^9-8 z^5 a^9+5 z^3 a^9-z a^9+8 z^8 a^8-17 z^6 a^8+12 z^4 a^8-5 z^2 a^8+a^8+8 z^9 a^7-11 z^7 a^7-z^5 a^7+z^3 a^7+z a^7+3 z^{10} a^6+14 z^8 a^6-45 z^6 a^6+39 z^4 a^6-16 z^2 a^6+2 a^6+17 z^9 a^5-30 z^7 a^5+15 z^5 a^5-8 z^3 a^5+5 z a^5+3 z^{10} a^4+17 z^8 a^4-47 z^6 a^4+40 z^4 a^4-13 z^2 a^4+9 z^9 a^3-7 z^7 a^3-3 z^5 a^3+z^3 a^3+3 z a^3+11 z^8 a^2-16 z^6 a^2+10 z^4 a^2-z^2 a^2-2 a^2+8 z^7 a-10 z^5 a+4 z^3 a+4 z^6-5 z^4+2 z^2+z^5 a^{-1} -z^3 a^{-1}
The A2 invariant Data:K11a273/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a273/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: (0, 2)
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
0 16 0 -80 -16 0 \frac{640}{3} -\frac{32}{3} 80 0 128 0 0 -136 144 -\frac{224}{3} -\frac{296}{3} 8

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 K11a273. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-8-7-6-5-4-3-2-10123χ
5           1-1
3          3 3
1         61 -5
-1        113  8
-3       127   -5
-5      1410    4
-7     1212     0
-9    1014      -4
-11   712       5
-13  310        -7
-15 17         6
-17 3          -3
-191           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-8 {\mathbb Z}
r=-7 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-5 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-4 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-3 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=-2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{14}
r=-1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{11}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\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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K11a272.gif

K11a272

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K11a274