K11a270

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K11a269.gif

K11a269

K11a271.gif

K11a271

Contents

K11a270.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a270 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X10,3,11,4 X12,6,13,5 X20,8,21,7 X18,10,19,9 X16,11,17,12 X22,13,1,14 X4,16,5,15 X2,17,3,18 X8,20,9,19 X14,21,15,22
Gauss code 1, -9, 2, -8, 3, -1, 4, -10, 5, -2, 6, -3, 7, -11, 8, -6, 9, -5, 10, -4, 11, -7
Dowker-Thistlethwaite code 6 10 12 20 18 16 22 4 2 8 14
A Braid Representative
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A Morse Link Presentation K11a270 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:K11a270/ThurstonBennequinNumber
Hyperbolic Volume 17.0324
A-Polynomial See Data:K11a270/A-polynomial

[edit Notes for K11a270'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 K11a270's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+12 t^2-32 t+45-32 t^{-1} +12 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6-2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 137, 0 }
Jones polynomial q^6-4 q^5+9 q^4-14 q^3+19 q^2-22 q+22-19 q^{-1} +14 q^{-2} -8 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} -z^4-a^4 z^2+2 a^2 z^2-3 z^2 a^{-2} +z^2 a^{-4} -z^2+a^2- a^{-2} + a^{-4}
Kauffman polynomial (db, data sources) 3 z^{10} a^{-2} +3 z^{10}+7 a z^9+15 z^9 a^{-1} +8 z^9 a^{-3} +8 a^2 z^8+7 z^8 a^{-2} +8 z^8 a^{-4} +7 z^8+7 a^3 z^7-8 a z^7-39 z^7 a^{-1} -20 z^7 a^{-3} +4 z^7 a^{-5} +4 a^4 z^6-9 a^2 z^6-35 z^6 a^{-2} -22 z^6 a^{-4} +z^6 a^{-6} -25 z^6+a^5 z^5-10 a^3 z^5+3 a z^5+40 z^5 a^{-1} +17 z^5 a^{-3} -9 z^5 a^{-5} -6 a^4 z^4-a^2 z^4+39 z^4 a^{-2} +18 z^4 a^{-4} -2 z^4 a^{-6} +24 z^4-a^5 z^3+3 a^3 z^3-a z^3-16 z^3 a^{-1} -8 z^3 a^{-3} +3 z^3 a^{-5} +2 a^4 z^2+3 a^2 z^2-14 z^2 a^{-2} -7 z^2 a^{-4} -6 z^2+2 z a^{-1} +2 z a^{-3} -a^2+ a^{-2} + a^{-4}
The A2 invariant Data:K11a270/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a270/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a80,}

Vassiliev invariants

V2 and V3: (-2, -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
-8 -8 32 \frac{116}{3} \frac{52}{3} 64 \frac{304}{3} \frac{256}{3} -40 -\frac{256}{3} 32 -\frac{928}{3} -\frac{416}{3} -\frac{2791}{15} \frac{2324}{15} -\frac{15844}{45} \frac{631}{9} -\frac{1351}{15}

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 K11a270. 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        83  -5
5       116   5
3      118    -3
1     1111     0
-1    912      3
-3   510       -5
-5  39        6
-7 15         -4
-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}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{11}
r=1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
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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K11a269.gif

K11a269

K11a271.gif

K11a271