K11a332

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K11a331.gif

K11a331

K11a333.gif

K11a333

Contents

K11a332.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a332 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X14,3,15,4 X10,6,11,5 X18,7,19,8 X2,10,3,9 X22,11,1,12 X20,14,21,13 X4,15,5,16 X12,18,13,17 X8,19,9,20 X16,22,17,21
Gauss code 1, -5, 2, -8, 3, -1, 4, -10, 5, -3, 6, -9, 7, -2, 8, -11, 9, -4, 10, -7, 11, -6
Dowker-Thistlethwaite code 6 14 10 18 2 22 20 4 12 8 16
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation K11a332 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a332/ThurstonBennequinNumber
Hyperbolic Volume 19.712
A-Polynomial See Data:K11a332/A-polynomial

[edit Notes for K11a332's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus [2,4]
Rasmussen s-Invariant 0

[edit Notes for K11a332's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-7 t^3+22 t^2-40 t+49-40 t^{-1} +22 t^{-2} -7 t^{-3} + t^{-4}
Conway polynomial z^8+z^6+z^2+1
2nd Alexander ideal (db, data sources) \left\{t^2-t+1\right\}
Determinant and Signature { 189, 0 }
Jones polynomial q^6-4 q^5+10 q^4-19 q^3+26 q^2-30 q+32-27 q^{-1} +21 q^{-2} -13 q^{-3} +5 q^{-4} - q^{-5}
HOMFLY-PT polynomial (db, data sources) z^8-a^2 z^6-2 z^6 a^{-2} +4 z^6-2 a^2 z^4-6 z^4 a^{-2} +z^4 a^{-4} +7 z^4-2 a^2 z^2-7 z^2 a^{-2} +2 z^2 a^{-4} +8 z^2-2 a^2-4 a^{-2} + a^{-4} +6
Kauffman polynomial (db, data sources) 4 z^{10} a^{-2} +4 z^{10}+14 a z^9+23 z^9 a^{-1} +9 z^9 a^{-3} +19 a^2 z^8+17 z^8 a^{-2} +8 z^8 a^{-4} +28 z^8+13 a^3 z^7-8 a z^7-35 z^7 a^{-1} -10 z^7 a^{-3} +4 z^7 a^{-5} +5 a^4 z^6-30 a^2 z^6-53 z^6 a^{-2} -15 z^6 a^{-4} +z^6 a^{-6} -72 z^6+a^5 z^5-15 a^3 z^5-18 a z^5-6 z^5 a^{-3} -8 z^5 a^{-5} -2 a^4 z^4+17 a^2 z^4+44 z^4 a^{-2} +11 z^4 a^{-4} -2 z^4 a^{-6} +50 z^4+6 a^3 z^3+14 a z^3+15 z^3 a^{-1} +13 z^3 a^{-3} +6 z^3 a^{-5} -6 a^2 z^2-16 z^2 a^{-2} -4 z^2 a^{-4} +z^2 a^{-6} -17 z^2-2 a^3 z-5 a z-7 z a^{-1} -6 z a^{-3} -2 z a^{-5} +2 a^2+4 a^{-2} + a^{-4} +6
The A2 invariant -q^{14}+3 q^{12}-5 q^{10}+2 q^8-4 q^4+9 q^2-3+7 q^{-2} -2 q^{-4} -3 q^{-6} +3 q^{-8} -6 q^{-10} +3 q^{-12} - q^{-16} + q^{-18}
The G2 invariant Data:K11a332/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{14}{3} \frac{10}{3} -32 -\frac{80}{3} \frac{64}{3} -8 \frac{32}{3} 32 \frac{56}{3} \frac{40}{3} \frac{2911}{30} -\frac{262}{15} \frac{5222}{45} -\frac{415}{18} \frac{511}{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 K11a332. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-5-4-3-2-10123456χ
13           11
11          3 -3
9         71 6
7        123  -9
5       147   7
3      1612    -4
1     1614     2
-1    1217      5
-3   915       -6
-5  412        8
-7 19         -8
-9 4          4
-111           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-1 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=0 {\mathbb Z}^{17}\oplus{\mathbb Z}_2^{15} {\mathbb Z}^{16}
r=1 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{16} {\mathbb Z}^{16}
r=2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{14}
r=3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=6 {\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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K11a331.gif

K11a331

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K11a333