K11a295

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K11a294.gif

K11a294

K11a296.gif

K11a296

Contents

K11a295.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a295 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X10,3,11,4 X16,5,17,6 X22,8,1,7 X18,9,19,10 X2,11,3,12 X20,13,21,14 X8,15,9,16 X4,17,5,18 X12,19,13,20 X14,21,15,22
Gauss code 1, -6, 2, -9, 3, -1, 4, -8, 5, -2, 6, -10, 7, -11, 8, -3, 9, -5, 10, -7, 11, -4
Dowker-Thistlethwaite code 6 10 16 22 18 2 20 8 4 12 14
A Braid Representative
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A Morse Link Presentation K11a295 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a295/ThurstonBennequinNumber
Hyperbolic Volume 15.5092
A-Polynomial See Data:K11a295/A-polynomial

[edit Notes for K11a295's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a295's four dimensional invariants]

Polynomial invariants

Alexander polynomial -3 t^3+14 t^2-24 t+27-24 t^{-1} +14 t^{-2} -3 t^{-3}
Conway polynomial -3 z^6-4 z^4+5 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 109, -4 }
Jones polynomial 1-3 q^{-1} +7 q^{-2} -11 q^{-3} +15 q^{-4} -17 q^{-5} +18 q^{-6} -15 q^{-7} +11 q^{-8} -7 q^{-9} +3 q^{-10} - q^{-11}
HOMFLY-PT polynomial (db, data sources) -z^2 a^{10}-2 a^{10}+3 z^4 a^8+8 z^2 a^8+4 a^8-2 z^6 a^6-7 z^4 a^6-8 z^2 a^6-4 a^6-z^6 a^4-z^4 a^4+4 z^2 a^4+3 a^4+z^4 a^2+2 z^2 a^2
Kauffman polynomial (db, data sources) z^5 a^{13}-2 z^3 a^{13}+z a^{13}+3 z^6 a^{12}-5 z^4 a^{12}+z^2 a^{12}+5 z^7 a^{11}-8 z^5 a^{11}+3 z^3 a^{11}-2 z a^{11}+6 z^8 a^{10}-11 z^6 a^{10}+10 z^4 a^{10}-7 z^2 a^{10}+2 a^{10}+5 z^9 a^9-8 z^7 a^9+5 z^5 a^9+2 z^3 a^9-z a^9+2 z^{10} a^8+5 z^8 a^8-24 z^6 a^8+36 z^4 a^8-18 z^2 a^8+4 a^8+10 z^9 a^7-29 z^7 a^7+35 z^5 a^7-16 z^3 a^7+3 z a^7+2 z^{10} a^6+4 z^8 a^6-25 z^6 a^6+37 z^4 a^6-22 z^2 a^6+4 a^6+5 z^9 a^5-13 z^7 a^5+13 z^5 a^5-9 z^3 a^5+z a^5+5 z^8 a^4-14 z^6 a^4+13 z^4 a^4-10 z^2 a^4+3 a^4+3 z^7 a^3-8 z^5 a^3+4 z^3 a^3+z^6 a^2-3 z^4 a^2+2 z^2 a^2
The A2 invariant Data:K11a295/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a295/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a120,}

Vassiliev invariants

V2 and V3: (5, -13)
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
20 -104 200 \frac{1990}{3} \frac{386}{3} -2080 -\frac{13424}{3} -\frac{2336}{3} -808 \frac{4000}{3} 5408 \frac{39800}{3} \frac{7720}{3} \frac{182239}{6} -\frac{3686}{3} \frac{134198}{9} \frac{6469}{18} \frac{12895}{6}

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 K11a295. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-9-8-7-6-5-4-3-2-1012χ
1           11
-1          2 -2
-3         51 4
-5        73  -4
-7       84   4
-9      97    -2
-11     98     1
-13    69      3
-15   59       -4
-17  26        4
-19 15         -4
-21 2          2
-231           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-5 i=-3
r=-9 {\mathbb Z}
r=-8 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-6 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-5 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-4 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\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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K11a294.gif

K11a294

K11a296.gif

K11a296