K11a77

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K11a76.gif

K11a76

K11a78.gif

K11a78

Contents

K11a77.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a77 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X12,6,13,5 X14,7,15,8 X18,9,19,10 X2,11,3,12 X22,14,1,13 X20,15,21,16 X8,17,9,18 X16,19,17,20 X6,21,7,22
Gauss code 1, -6, 2, -1, 3, -11, 4, -9, 5, -2, 6, -3, 7, -4, 8, -10, 9, -5, 10, -8, 11, -7
Dowker-Thistlethwaite code 4 10 12 14 18 2 22 20 8 16 6
A Braid Representative
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A Morse Link Presentation K11a77 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 3
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a77/ThurstonBennequinNumber
Hyperbolic Volume 15.2573
A-Polynomial See Data:K11a77/A-polynomial

[edit Notes for K11a77'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 K11a77's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-12 t^2+31 t-41+31 t^{-1} -12 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 131, -2 }
Jones polynomial -q^2+4 q-8+14 q^{-1} -18 q^{-2} +21 q^{-3} -21 q^{-4} +18 q^{-5} -13 q^{-6} +8 q^{-7} -4 q^{-8} + q^{-9}
HOMFLY-PT polynomial (db, data sources) z^2 a^8-2 z^4 a^6-2 z^2 a^6+z^6 a^4+z^4 a^4-a^4+z^6 a^2+2 z^4 a^2+3 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^9-10 z^5 a^9+7 z^3 a^9-z a^9+6 z^8 a^8-13 z^6 a^8+7 z^4 a^8+4 z^9 a^7+2 z^7 a^7-21 z^5 a^7+18 z^3 a^7-4 z a^7+z^{10} a^6+14 z^8 a^6-33 z^6 a^6+20 z^4 a^6-3 z^2 a^6+8 z^9 a^5-26 z^5 a^5+22 z^3 a^5-5 z a^5+z^{10} a^4+15 z^8 a^4-27 z^6 a^4+10 z^4 a^4+2 z^2 a^4-a^4+4 z^9 a^3+9 z^7 a^3-26 z^5 a^3+17 z^3 a^3-3 z a^3+7 z^8 a^2-4 z^6 a^2-7 z^4 a^2+7 z^2 a^2-2 a^2+7 z^7 a-10 z^5 a+5 z^3 a-z a+4 z^6-6 z^4+3 z^2+z^5 a^{-1} -z^3 a^{-1}
The A2 invariant Data:K11a77/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a77/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, -1)
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 -8 8 \frac{62}{3} \frac{10}{3} -32 -\frac{272}{3} -\frac{128}{3} -8 \frac{32}{3} 32 \frac{248}{3} \frac{40}{3} \frac{9871}{30} -\frac{662}{15} \frac{13142}{45} -\frac{271}{18} \frac{1711}{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=-2 is the signature of K11a77. 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         51 -4
-1        93  6
-3       106   -4
-5      118    3
-7     1010     0
-9    811      -3
-11   510       5
-13  38        -5
-15 15         4
-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}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-5 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-4 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-3 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=-1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
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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K11a76.gif

K11a76

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K11a78