K11a84

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K11a83.gif

K11a83

K11a85.gif

K11a85

Contents

K11a84.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a84 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X12,5,13,6 X16,8,17,7 X2,10,3,9 X22,11,1,12 X20,13,21,14 X18,15,19,16 X6,18,7,17 X14,19,15,20 X8,21,9,22
Gauss code 1, -5, 2, -1, 3, -9, 4, -11, 5, -2, 6, -3, 7, -10, 8, -4, 9, -8, 10, -7, 11, -6
Dowker-Thistlethwaite code 4 10 12 16 2 22 20 18 6 14 8
A Braid Representative
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A Morse Link Presentation K11a84 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a84/ThurstonBennequinNumber
Hyperbolic Volume 13.8696
A-Polynomial See Data:K11a84/A-polynomial

[edit Notes for K11a84'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 K11a84's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+10 t^2-23 t+31-23 t^{-1} +10 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6-2 z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 101, 0 }
Jones polynomial -q^5+3 q^4-6 q^3+11 q^2-14 q+16-16 q^{-1} +14 q^{-2} -10 q^{-3} +6 q^{-4} -3 q^{-5} + q^{-6}
HOMFLY-PT polynomial (db, data sources) -a^2 z^6-z^6+a^4 z^4-3 a^2 z^4+2 z^4 a^{-2} -2 z^4+2 a^4 z^2-4 a^2 z^2+4 z^2 a^{-2} -z^2 a^{-4} -2 z^2+a^4-a^2+3 a^{-2} - a^{-4} -1
Kauffman polynomial (db, data sources) a^2 z^{10}+z^{10}+3 a^3 z^9+6 a z^9+3 z^9 a^{-1} +4 a^4 z^8+6 a^2 z^8+4 z^8 a^{-2} +6 z^8+3 a^5 z^7-3 a^3 z^7-10 a z^7+4 z^7 a^{-3} +a^6 z^6-10 a^4 z^6-18 a^2 z^6-z^6 a^{-2} +3 z^6 a^{-4} -11 z^6-9 a^5 z^5-4 a^3 z^5+9 a z^5-z^5 a^{-1} -4 z^5 a^{-3} +z^5 a^{-5} -3 a^6 z^4+7 a^4 z^4+18 a^2 z^4-8 z^4 a^{-2} -6 z^4 a^{-4} +6 z^4+7 a^5 z^3+3 a^3 z^3-7 a z^3-3 z^3 a^{-1} -2 z^3 a^{-3} -2 z^3 a^{-5} +2 a^6 z^2-3 a^4 z^2-9 a^2 z^2+9 z^2 a^{-2} +4 z^2 a^{-4} +z^2-a^5 z+2 a z+2 z a^{-1} +2 z a^{-3} +z a^{-5} +a^4+a^2-3 a^{-2} - a^{-4} -1
The A2 invariant Data:K11a84/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a84/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_119,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a88,}

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{82}{3} \frac{62}{3} -64 -\frac{416}{3} -\frac{224}{3} -16 -\frac{32}{3} 128 -\frac{328}{3} -\frac{248}{3} \frac{4769}{30} \frac{3662}{15} -\frac{11342}{45} \frac{991}{18} -\frac{2911}{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 K11a84. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-6-5-4-3-2-1012345χ
11           1-1
9          2 2
7         41 -3
5        72  5
3       74   -3
1      97    2
-1     88     0
-3    68      -2
-5   48       4
-7  26        -4
-9 14         3
-11 2          -2
-131           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=5 {\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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K11a83.gif

K11a83

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K11a85