K11n76

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K11n75.gif

K11n75

K11n77.gif

K11n77

Contents

K11n76.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n76 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X14,5,15,6 X2837 X9,20,10,21 X16,11,17,12 X18,13,19,14 X6,15,7,16 X12,17,13,18 X19,22,20,1 X21,10,22,11
Gauss code 1, -4, 2, -1, 3, -8, 4, -2, -5, 11, 6, -9, 7, -3, 8, -6, 9, -7, -10, 5, -11, 10
Dowker-Thistlethwaite code 4 8 14 2 -20 16 18 6 12 -22 -10
A Braid Representative
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A Morse Link Presentation K11n76 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:K11n76/ThurstonBennequinNumber
Hyperbolic Volume 12.3125
A-Polynomial See Data:K11n76/A-polynomial

[edit Notes for K11n76's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 2
Rasmussen s-Invariant 4

[edit Notes for K11n76's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-3 t^3+6 t^2-8 t+9-8 t^{-1} +6 t^{-2} -3 t^{-3} + t^{-4}
Conway polynomial z^8+5 z^6+8 z^4+5 z^2+1
2nd Alexander ideal (db, data sources) \left\{t^2-t+1\right\}
Determinant and Signature { 45, -4 }
Jones polynomial -q+2-4 q^{-1} +7 q^{-2} -6 q^{-3} +8 q^{-4} -7 q^{-5} +5 q^{-6} -4 q^{-7} + q^{-8}
HOMFLY-PT polynomial (db, data sources) z^2 a^8+2 a^8-z^6 a^6-6 z^4 a^6-13 z^2 a^6-10 a^6+z^8 a^4+7 z^6 a^4+19 z^4 a^4+25 z^2 a^4+13 a^4-z^6 a^2-5 z^4 a^2-8 z^2 a^2-4 a^2
Kauffman polynomial (db, data sources) z^2 a^{10}+4 z^3 a^9-2 z a^9+2 z^6 a^8-2 z^4 a^8+z^2 a^8+2 a^8+5 z^7 a^7-18 z^5 a^7+26 z^3 a^7-14 z a^7+4 z^8 a^6-13 z^6 a^6+16 z^4 a^6-17 z^2 a^6+10 a^6+z^9 a^5+5 z^7 a^5-31 z^5 a^5+42 z^3 a^5-21 z a^5+6 z^8 a^4-24 z^6 a^4+31 z^4 a^4-26 z^2 a^4+13 a^4+z^9 a^3+z^7 a^3-18 z^5 a^3+28 z^3 a^3-13 z a^3+2 z^8 a^2-9 z^6 a^2+13 z^4 a^2-9 z^2 a^2+4 a^2+z^7 a-5 z^5 a+8 z^3 a-4 z a
The A2 invariant q^{28}+q^{24}-3 q^{22}-3 q^{20}-3 q^{18}-2 q^{16}+4 q^{14}+2 q^{12}+6 q^{10}+2 q^8+q^6-2 q^2- q^{-2}
The G2 invariant Data:K11n76/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_62, K11n78,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11n78,}

Vassiliev invariants

V2 and V3: (5, -7)
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 -56 200 \frac{886}{3} \frac{98}{3} -1120 -\frac{4496}{3} -\frac{704}{3} -152 \frac{4000}{3} 1568 \frac{17720}{3} \frac{1960}{3} \frac{50767}{6} \frac{2090}{3} \frac{21446}{9} \frac{853}{18} \frac{1423}{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 K11n76. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-6-5-4-3-2-10123χ
3         1-1
1        1 1
-1       31 -2
-3      41  3
-5     34   1
-7    53    2
-9   23     1
-11  35      -2
-13 12       1
-15 3        -3
-171         1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-5 i=-3
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
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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K11n75.gif

K11n75

K11n77.gif

K11n77