K11a30

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K11a29.gif

K11a29

K11a31.gif

K11a31

Contents

K11a30.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a30 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8394 X12,6,13,5 X18,8,19,7 X2,9,3,10 X16,12,17,11 X20,13,21,14 X6,16,7,15 X10,18,11,17 X22,19,1,20 X14,21,15,22
Gauss code 1, -5, 2, -1, 3, -8, 4, -2, 5, -9, 6, -3, 7, -11, 8, -6, 9, -4, 10, -7, 11, -10
Dowker-Thistlethwaite code 4 8 12 18 2 16 20 6 10 22 14
A Braid Representative
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A Morse Link Presentation K11a30 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:K11a30/ThurstonBennequinNumber
Hyperbolic Volume 17.3478
A-Polynomial See Data:K11a30/A-polynomial

[edit Notes for K11a30's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 0

[edit Notes for K11a30's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+13 t^2-35 t+49-35 t^{-1} +13 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6+z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 149, 0 }
Jones polynomial q^6-4 q^5+9 q^4-15 q^3+21 q^2-24 q+24-21 q^{-1} +16 q^{-2} -9 q^{-3} +4 q^{-4} - q^{-5}
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-2} -z^6+2 a^2 z^4-2 z^4 a^{-2} +z^4 a^{-4} -a^4 z^2+a^2 z^2-4 z^2 a^{-2} +z^2 a^{-4} +2 z^2-2 a^{-2} + a^{-4} +2
Kauffman polynomial (db, data sources) 2 z^{10} a^{-2} +2 z^{10}+7 a z^9+13 z^9 a^{-1} +6 z^9 a^{-3} +10 a^2 z^8+15 z^8 a^{-2} +7 z^8 a^{-4} +18 z^8+8 a^3 z^7+a z^7-15 z^7 a^{-1} -4 z^7 a^{-3} +4 z^7 a^{-5} +4 a^4 z^6-13 a^2 z^6-42 z^6 a^{-2} -15 z^6 a^{-4} +z^6 a^{-6} -43 z^6+a^5 z^5-11 a^3 z^5-17 a z^5-8 z^5 a^{-1} -12 z^5 a^{-3} -9 z^5 a^{-5} -5 a^4 z^4+6 a^2 z^4+34 z^4 a^{-2} +10 z^4 a^{-4} -2 z^4 a^{-6} +33 z^4-a^5 z^3+6 a^3 z^3+13 a z^3+11 z^3 a^{-1} +11 z^3 a^{-3} +6 z^3 a^{-5} +2 a^4 z^2-a^2 z^2-13 z^2 a^{-2} -3 z^2 a^{-4} +z^2 a^{-6} -12 z^2-a^3 z-3 a z-3 z a^{-1} -2 z a^{-3} -z a^{-5} +2 a^{-2} + a^{-4} +2
The A2 invariant Data:K11a30/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a30/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a272,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a189, K11a272,}

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{208}{3} \frac{160}{3} -8 -\frac{32}{3} 32 \frac{56}{3} \frac{40}{3} \frac{3809}{30} \frac{1462}{15} -\frac{902}{45} \frac{127}{18} -\frac{991}{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 K11a30. 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         61 5
7        93  -6
5       126   6
3      129    -3
1     1212     0
-1    1013      3
-3   611       -5
-5  310        7
-7 16         -5
-9 3          3
-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}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=0 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{12}
r=1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
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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K11a29.gif

K11a29

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K11a31