K11a297

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K11a296.gif

K11a296

K11a298.gif

K11a298

Contents

K11a297.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

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Knot presentations

Planar diagram presentation X6271 X10,3,11,4 X16,5,17,6 X22,8,1,7 X20,9,21,10 X18,12,19,11 X2,13,3,14 X8,15,9,16 X12,18,13,17 X4,19,5,20 X14,21,15,22
Gauss code 1, -7, 2, -10, 3, -1, 4, -8, 5, -2, 6, -9, 7, -11, 8, -3, 9, -6, 10, -5, 11, -4
Dowker-Thistlethwaite code 6 10 16 22 20 18 2 8 12 4 14
A Braid Representative
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A Morse Link Presentation K11a297 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a297/ThurstonBennequinNumber
Hyperbolic Volume 18.8513
A-Polynomial See Data:K11a297/A-polynomial

[edit Notes for K11a297's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus [1,3]
Rasmussen s-Invariant 2

[edit Notes for K11a297's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-15 t^2+42 t-57+42 t^{-1} -15 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6-3 z^4+1
2nd Alexander ideal (db, data sources) \left\{t^2-3 t+1\right\}
Determinant and Signature { 175, -2 }
Jones polynomial q^3-4 q^2+10 q-17+24 q^{-1} -28 q^{-2} +29 q^{-3} -25 q^{-4} +19 q^{-5} -12 q^{-6} +5 q^{-7} - q^{-8}
HOMFLY-PT polynomial (db, data sources) -z^4 a^6-a^6+z^6 a^4+z^4 a^4+3 z^2 a^4+3 a^4+z^6 a^2-z^4 a^2-4 z^2 a^2-3 a^2-2 z^4+2+z^2 a^{-2}
Kauffman polynomial (db, data sources) 4 a^4 z^{10}+4 a^2 z^{10}+12 a^5 z^9+21 a^3 z^9+9 a z^9+16 a^6 z^8+19 a^4 z^8+11 a^2 z^8+8 z^8+12 a^7 z^7-9 a^5 z^7-41 a^3 z^7-16 a z^7+4 z^7 a^{-1} +5 a^8 z^6-24 a^6 z^6-51 a^4 z^6-40 a^2 z^6+z^6 a^{-2} -17 z^6+a^9 z^5-15 a^7 z^5-7 a^5 z^5+27 a^3 z^5+10 a z^5-8 z^5 a^{-1} -3 a^8 z^4+11 a^6 z^4+37 a^4 z^4+38 a^2 z^4-2 z^4 a^{-2} +13 z^4+4 a^7 z^3+a^5 z^3-12 a^3 z^3-5 a z^3+4 z^3 a^{-1} -3 a^6 z^2-13 a^4 z^2-17 a^2 z^2+z^2 a^{-2} -6 z^2+2 a^5 z+4 a^3 z+2 a z+a^6+3 a^4+3 a^2+2
The A2 invariant Data:K11a297/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a297/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a125,}

Vassiliev invariants

V2 and V3: (0, -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
0 -8 0 48 24 0 -\frac{368}{3} \frac{64}{3} -104 0 32 0 0 264 -392 520 56 104

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=-2 is the signature of K11a297. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-7-6-5-4-3-2-101234χ
7           11
5          3 -3
3         71 6
1        103  -7
-1       147   7
-3      1511    -4
-5     1413     1
-7    1115      4
-9   814       -6
-11  411        7
-13 18         -7
-15 4          4
-171           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-4 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-3 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=-2 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{14}
r=-1 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{15} {\mathbb Z}^{15}
r=0 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{14}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
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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K11a296.gif

K11a296

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K11a298