K11a265

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K11a264.gif

K11a264

K11a266.gif

K11a266

Contents

K11a265.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a265 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X8394 X16,6,17,5 X14,7,15,8 X4,9,5,10 X20,12,21,11 X18,14,19,13 X2,16,3,15 X22,17,1,18 X12,20,13,19 X10,22,11,21
Gauss code 1, -8, 2, -5, 3, -1, 4, -2, 5, -11, 6, -10, 7, -4, 8, -3, 9, -7, 10, -6, 11, -9
Dowker-Thistlethwaite code 6 8 16 14 4 20 18 2 22 12 10
A Braid Representative
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A Morse Link Presentation K11a265 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:K11a265/ThurstonBennequinNumber
Hyperbolic Volume 14.8555
A-Polynomial See Data:K11a265/A-polynomial

[edit Notes for K11a265's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 0

[edit Notes for K11a265's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+11 t^2-25 t+33-25 t^{-1} +11 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6-z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 109, 0 }
Jones polynomial -q^7+3 q^6-6 q^5+10 q^4-14 q^3+17 q^2-17 q+16-12 q^{-1} +8 q^{-2} -4 q^{-3} + q^{-4}
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-2} -z^6+a^2 z^4-2 z^4 a^{-2} +2 z^4 a^{-4} -2 z^4+a^2 z^2-2 z^2 a^{-2} +4 z^2 a^{-4} -z^2 a^{-6} -z^2- a^{-2} +2 a^{-4} - a^{-6} +1
Kauffman polynomial (db, data sources) 2 z^{10} a^{-2} +2 z^{10} a^{-4} +6 z^9 a^{-1} +10 z^9 a^{-3} +4 z^9 a^{-5} +7 z^8 a^{-2} +z^8 a^{-4} +3 z^8 a^{-6} +9 z^8+10 a z^7-6 z^7 a^{-1} -31 z^7 a^{-3} -14 z^7 a^{-5} +z^7 a^{-7} +8 a^2 z^6-29 z^6 a^{-2} -22 z^6 a^{-4} -12 z^6 a^{-6} -11 z^6+4 a^3 z^5-12 a z^5-3 z^5 a^{-1} +30 z^5 a^{-3} +13 z^5 a^{-5} -4 z^5 a^{-7} +a^4 z^4-8 a^2 z^4+27 z^4 a^{-2} +31 z^4 a^{-4} +14 z^4 a^{-6} +z^4-2 a^3 z^3+3 a z^3-12 z^3 a^{-3} -3 z^3 a^{-5} +4 z^3 a^{-7} +2 a^2 z^2-10 z^2 a^{-2} -14 z^2 a^{-4} -6 z^2 a^{-6} +z a^{-1} +2 z a^{-3} -z a^{-7} + a^{-2} +2 a^{-4} + a^{-6} +1
The A2 invariant Data:K11a265/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a265/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a56, K11a185,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a185,}

Vassiliev invariants

V2 and V3: (1, 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
4 16 8 \frac{158}{3} \frac{34}{3} 64 \frac{544}{3} -\frac{128}{3} 112 \frac{32}{3} 128 \frac{632}{3} \frac{136}{3} \frac{19471}{30} -\frac{1594}{5} \frac{23102}{45} \frac{2129}{18} \frac{2671}{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=0 is the signature of K11a265. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234567χ
15           1-1
13          2 2
11         41 -3
9        62  4
7       84   -4
5      96    3
3     88     0
1    89      -1
-1   59       4
-3  37        -4
-5 15         4
-7 3          -3
-91           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=7 {\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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K11a264.gif

K11a264

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K11a266