K11n81

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K11n80.gif

K11n80

K11n82.gif

K11n82

Contents

K11n81.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n81 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X5,16,6,17 X2837 X9,21,10,20 X11,1,12,22 X13,19,14,18 X15,6,16,7 X17,13,18,12 X19,15,20,14 X21,11,22,10
Gauss code 1, -4, 2, -1, -3, 8, 4, -2, -5, 11, -6, 9, -7, 10, -8, 3, -9, 7, -10, 5, -11, 6
Dowker-Thistlethwaite code 4 8 -16 2 -20 -22 -18 -6 -12 -14 -10
A Braid Representative
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A Morse Link Presentation K11n81 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 3
3-genus 4
Bridge index 4
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n81/ThurstonBennequinNumber
Hyperbolic Volume 10.956
A-Polynomial See Data:K11n81/A-polynomial

[edit Notes for K11n81's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n81's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+3 t^3-4 t^2+4 t-3+4 t^{-1} -4 t^{-2} +3 t^{-3} - t^{-4}
Conway polynomial -z^8-5 z^6-6 z^4-z^2+1
2nd Alexander ideal (db, data sources) \left\{t^2-t+1\right\}
Determinant and Signature { 27, 6 }
Jones polynomial 2 q^9-3 q^8+3 q^7-5 q^6+4 q^5-4 q^4+4 q^3-q^2+q
HOMFLY-PT polynomial (db, data sources) -z^8 a^{-6} +z^6 a^{-4} -7 z^6 a^{-6} +z^6 a^{-8} +6 z^4 a^{-4} -18 z^4 a^{-6} +6 z^4 a^{-8} +12 z^2 a^{-4} -23 z^2 a^{-6} +11 z^2 a^{-8} -z^2 a^{-10} +8 a^{-4} -13 a^{-6} +7 a^{-8} - a^{-10}
Kauffman polynomial (db, data sources) z^9 a^{-5} +z^9 a^{-7} +z^8 a^{-4} +5 z^8 a^{-6} +4 z^8 a^{-8} -4 z^7 a^{-5} +2 z^7 a^{-7} +6 z^7 a^{-9} -7 z^6 a^{-4} -28 z^6 a^{-6} -18 z^6 a^{-8} +3 z^6 a^{-10} -30 z^5 a^{-7} -30 z^5 a^{-9} +18 z^4 a^{-4} +50 z^4 a^{-6} +20 z^4 a^{-8} -12 z^4 a^{-10} +13 z^3 a^{-5} +53 z^3 a^{-7} +41 z^3 a^{-9} +z^3 a^{-11} -20 z^2 a^{-4} -38 z^2 a^{-6} -10 z^2 a^{-8} +8 z^2 a^{-10} -11 z a^{-5} -26 z a^{-7} -18 z a^{-9} -3 z a^{-11} +8 a^{-4} +13 a^{-6} +7 a^{-8} + a^{-10}
The A2 invariant  q^{-4} + q^{-6} +3 q^{-8} +3 q^{-10} +2 q^{-12} -4 q^{-16} -3 q^{-18} -5 q^{-20} + q^{-24} +2 q^{-26} +2 q^{-28} + q^{-32} - q^{-34}
The G2 invariant Data:K11n81/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: (-1, -6)
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 -48 8 -\frac{302}{3} \frac{158}{3} 192 416 256 272 -\frac{32}{3} 1152 \frac{1208}{3} -\frac{632}{3} \frac{123809}{30} \frac{7502}{15} \frac{83218}{45} \frac{6559}{18} -\frac{991}{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=6 is the signature of K11n81. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-10123456χ
19        22
17       1 -1
15      33 0
13     31  -2
11    131  -1
9   33    0
7  11     0
5 14      3
3         0
11        1
Integral Khovanov Homology

(db, data source)

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

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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K11n82