K11a67

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K11a66.gif

K11a66

K11a68.gif

K11a68

Contents

K11a67.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a67 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8394 X16,5,17,6 X12,8,13,7 X2,9,3,10 X18,12,19,11 X22,13,1,14 X20,15,21,16 X10,18,11,17 X6,19,7,20 X14,21,15,22
Gauss code 1, -5, 2, -1, 3, -10, 4, -2, 5, -9, 6, -4, 7, -11, 8, -3, 9, -6, 10, -8, 11, -7
Dowker-Thistlethwaite code 4 8 16 12 2 18 22 20 10 6 14
A Braid Representative
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A Morse Link Presentation K11a67 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a67/ThurstonBennequinNumber
Hyperbolic Volume 16.0998
A-Polynomial See Data:K11a67/A-polynomial

[edit Notes for K11a67's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus [1,3]
Rasmussen s-Invariant 0

[edit Notes for K11a67's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+12 t^2-29 t+39-29 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 { 125, 0 }
Jones polynomial q^4-4 q^3+9 q^2-14 q+18-20 q^{-1} +20 q^{-2} -16 q^{-3} +12 q^{-4} -7 q^{-5} +3 q^{-6} - q^{-7}
HOMFLY-PT polynomial (db, data sources) -z^2 a^6-a^6+2 z^4 a^4+3 z^2 a^4+a^4-z^6 a^2-z^4 a^2+z^2 a^2+2 a^2-z^6-2 z^4-3 z^2-2+z^4 a^{-2} +z^2 a^{-2} + a^{-2}
Kauffman polynomial (db, data sources) 2 a^4 z^{10}+2 a^2 z^{10}+4 a^5 z^9+11 a^3 z^9+7 a z^9+3 a^6 z^8+4 a^4 z^8+13 a^2 z^8+12 z^8+a^7 z^7-11 a^5 z^7-25 a^3 z^7+13 z^7 a^{-1} -11 a^6 z^6-27 a^4 z^6-40 a^2 z^6+9 z^6 a^{-2} -15 z^6-4 a^7 z^5+5 a^5 z^5+7 a^3 z^5-24 a z^5-18 z^5 a^{-1} +4 z^5 a^{-3} +13 a^6 z^4+29 a^4 z^4+24 a^2 z^4-8 z^4 a^{-2} +z^4 a^{-4} -z^4+5 a^7 z^3+5 a^5 z^3+8 a^3 z^3+18 a z^3+9 z^3 a^{-1} -z^3 a^{-3} -6 a^6 z^2-9 a^4 z^2+3 z^2 a^{-2} +6 z^2-2 a^7 z-3 a^5 z-3 a^3 z-4 a z-2 z a^{-1} +a^6+a^4-2 a^2- a^{-2} -2
The A2 invariant Data:K11a67/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a67/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a104, K11a168,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a317,}

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{158}{3} \frac{10}{3} -96 -208 32 -88 \frac{32}{3} 288 \frac{632}{3} \frac{40}{3} \frac{26671}{30} -\frac{2662}{15} \frac{18902}{45} \frac{2033}{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=0 is the signature of K11a67. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-7-6-5-4-3-2-101234χ
9           11
7          3 -3
5         61 5
3        83  -5
1       106   4
-1      119    -2
-3     99     0
-5    711      4
-7   59       -4
-9  27        5
-11 15         -4
-13 2          2
-151           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\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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K11a66

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K11a68