K11n36

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K11n35.gif

K11n35

K11n37.gif

K11n37

Contents

K11n36.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n36 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X5,13,6,12 X2837 X9,17,10,16 X11,18,12,19 X13,7,14,6 X15,20,16,21 X17,1,18,22 X19,14,20,15 X21,10,22,11
Gauss code 1, -4, 2, -1, -3, 7, 4, -2, -5, 11, -6, 3, -7, 10, -8, 5, -9, 6, -10, 8, -11, 9
Dowker-Thistlethwaite code 4 8 -12 2 -16 -18 -6 -20 -22 -14 -10
A Braid Representative
BraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gif
A Morse Link Presentation K11n36 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n36/ThurstonBennequinNumber
Hyperbolic Volume 14.4828
A-Polynomial See Data:K11n36/A-polynomial

[edit Notes for K11n36's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n36's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+4 t^3-8 t^2+13 t-15+13 t^{-1} -8 t^{-2} +4 t^{-3} - t^{-4}
Conway polynomial -z^8-4 z^6-4 z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 67, 2 }
Jones polynomial -2 q^6+5 q^5-8 q^4+11 q^3-11 q^2+11 q-9+6 q^{-1} -3 q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) -z^8 a^{-2} -6 z^6 a^{-2} +z^6 a^{-4} +z^6-13 z^4 a^{-2} +5 z^4 a^{-4} +4 z^4-12 z^2 a^{-2} +9 z^2 a^{-4} -z^2 a^{-6} +5 z^2-4 a^{-2} +5 a^{-4} -2 a^{-6} +2
Kauffman polynomial (db, data sources) 2 z^9 a^{-1} +2 z^9 a^{-3} +9 z^8 a^{-2} +5 z^8 a^{-4} +4 z^8+3 a z^7+z^7 a^{-1} +2 z^7 a^{-3} +4 z^7 a^{-5} +a^2 z^6-27 z^6 a^{-2} -14 z^6 a^{-4} +z^6 a^{-6} -11 z^6-9 a z^5-15 z^5 a^{-1} -14 z^5 a^{-3} -8 z^5 a^{-5} -3 a^2 z^4+30 z^4 a^{-2} +23 z^4 a^{-4} +4 z^4 a^{-6} +8 z^4+7 a z^3+14 z^3 a^{-1} +17 z^3 a^{-3} +13 z^3 a^{-5} +3 z^3 a^{-7} +2 a^2 z^2-18 z^2 a^{-2} -16 z^2 a^{-4} -5 z^2 a^{-6} -5 z^2-2 a z-5 z a^{-1} -6 z a^{-3} -6 z a^{-5} -3 z a^{-7} +4 a^{-2} +5 a^{-4} +2 a^{-6} +2
The A2 invariant q^8-q^6+2 q^4-q^2+ q^{-2} -3 q^{-4} +3 q^{-6} -2 q^{-8} +3 q^{-10} + q^{-12} +2 q^{-16} -2 q^{-18} - q^{-22}
The G2 invariant Data:K11n36/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n44,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11n7, K11n44,}

Vassiliev invariants

V2 and V3: (1, 3)
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 24 8 \frac{302}{3} \frac{106}{3} 96 304 64 56 \frac{32}{3} 288 \frac{1208}{3} \frac{424}{3} \frac{37951}{30} \frac{366}{5} \frac{22022}{45} \frac{1121}{18} \frac{1471}{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 K11n36. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-1012345χ
13         2-2
11        3 3
9       52 -3
7      63  3
5     55   0
3    66    0
1   46     2
-1  25      -3
-3 14       3
-5 2        -2
-71         1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=5 {\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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K11n35.gif

K11n35

K11n37.gif

K11n37