K11a318

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K11a317.gif

K11a317

K11a319.gif

K11a319

Contents

K11a318.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a318 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X12,4,13,3 X16,6,17,5 X22,8,1,7 X18,10,19,9 X2,12,3,11 X20,14,21,13 X4,16,5,15 X8,18,9,17 X14,20,15,19 X10,22,11,21
Gauss code 1, -6, 2, -8, 3, -1, 4, -9, 5, -11, 6, -2, 7, -10, 8, -3, 9, -5, 10, -7, 11, -4
Dowker-Thistlethwaite code 6 12 16 22 18 2 20 4 8 14 10
A Braid Representative
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A Morse Link Presentation K11a318 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 3
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a318/ThurstonBennequinNumber
Hyperbolic Volume 17.2773
A-Polynomial See Data:K11a318/A-polynomial

[edit Notes for K11a318's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -6

[edit Notes for K11a318's four dimensional invariants]

Polynomial invariants

Alexander polynomial 5 t^3-16 t^2+29 t-35+29 t^{-1} -16 t^{-2} +5 t^{-3}
Conway polynomial 5 z^6+14 z^4+10 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 135, 6 }
Jones polynomial -q^{14}+3 q^{13}-8 q^{12}+14 q^{11}-19 q^{10}+22 q^9-22 q^8+19 q^7-14 q^6+9 q^5-3 q^4+q^3
HOMFLY-PT polynomial (db, data sources) z^6 a^{-6} +3 z^6 a^{-8} +z^6 a^{-10} +3 z^4 a^{-6} +11 z^4 a^{-8} +z^4 a^{-10} -z^4 a^{-12} +3 z^2 a^{-6} +11 z^2 a^{-8} -2 z^2 a^{-10} -2 z^2 a^{-12} + a^{-6} +2 a^{-8} - a^{-10} - a^{-12}
Kauffman polynomial (db, data sources) 2 z^{10} a^{-10} +2 z^{10} a^{-12} +5 z^9 a^{-9} +11 z^9 a^{-11} +6 z^9 a^{-13} +6 z^8 a^{-8} +10 z^8 a^{-10} +12 z^8 a^{-12} +8 z^8 a^{-14} +3 z^7 a^{-7} -2 z^7 a^{-9} -13 z^7 a^{-11} -2 z^7 a^{-13} +6 z^7 a^{-15} +z^6 a^{-6} -14 z^6 a^{-8} -27 z^6 a^{-10} -28 z^6 a^{-12} -13 z^6 a^{-14} +3 z^6 a^{-16} -6 z^5 a^{-7} -15 z^5 a^{-9} -8 z^5 a^{-11} -9 z^5 a^{-13} -9 z^5 a^{-15} +z^5 a^{-17} -3 z^4 a^{-6} +14 z^4 a^{-8} +17 z^4 a^{-10} +17 z^4 a^{-12} +13 z^4 a^{-14} -4 z^4 a^{-16} +3 z^3 a^{-7} +14 z^3 a^{-9} +13 z^3 a^{-11} +11 z^3 a^{-13} +7 z^3 a^{-15} -2 z^3 a^{-17} +3 z^2 a^{-6} -10 z^2 a^{-8} -6 z^2 a^{-10} -6 z^2 a^{-14} +z^2 a^{-16} -4 z a^{-9} -3 z a^{-11} -3 z a^{-13} -3 z a^{-15} +z a^{-17} - a^{-6} +2 a^{-8} + a^{-10} - a^{-12}
The A2 invariant Data:K11a318/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a318/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: (10, 31)
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
40 248 800 \frac{5900}{3} \frac{844}{3} 9920 \frac{52112}{3} \frac{9056}{3} 2136 \frac{32000}{3} 30752 \frac{236000}{3} \frac{33760}{3} \frac{470575}{3} 6260 \frac{515716}{9} \frac{9493}{9} \frac{21631}{3}

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=6 is the signature of K11a318. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567891011χ
29           1-1
27          2 2
25         61 -5
23        82  6
21       116   -5
19      118    3
17     1111     0
15    811      -3
13   611       5
11  38        -5
9  6         6
713          -2
51           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=5 i=7
r=0 {\mathbb Z} {\mathbb Z}
r=1 {\mathbb Z}^{3}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=5 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=6 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=7 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=8 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=9 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=10 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=11 {\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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K11a317.gif

K11a317

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K11a319